Stability of Operators and Operator Semigroups [1st ed. 2010] 978-3-0346-0194-8, 978-3-0346-0195-5

Table of contents :
Front Matter ....Pages i-xi
Functional analytic tools (Tanja Eisner)....Pages 5-35
Stability of linear operators (Tanja Eisner)....Pages 37-77
Stability of C0-semigroups (Tanja Eisner)....Pages 79-132
Connections to ergodic and measure theory (Tanja Eisner)....Pages 133-162
Discrete vs. continuous (Tanja Eisner)....Pages 163-182
Back Matter ....Pages 183-204

Citation preview

Operator Theory: Advances and Applications

Vol. 209 Founded in 1979 by Israel Gohberg

Editors: Joseph A. Ball (Blacksburg, VA, USA) Harry Dym (Rehovot, Israel) Marin us A. Kaashoek (Amsterdam, The Netherlands) Heinz Langer (Vienna, Austria) Christiane Tretter (Bern, Switzerland) Associate Editors: Vadim Adamyan (Odessa, Ukraine) Albrecht Bottcher (Chemnitz, Germany) B. Malcolm Brown (Cardiff, UK) Raul Curto (Iowa City, lA, USA) Fritz Gesztesy (Columbia, MO, USA) Pavel Kurasov (Lund, Sweden) Leonid E. Lerer (Haifa, Israel) Vern Paulsen (Houston, TX, USA) Mihai Putinar (Santa Barbara, CA, USA) Leiba Rodman (Williamsburg, VI, USA) llya M. Spitkovsky (Williamsburg, VI, USA)

Subseries Linear Operators and Linear Systems Subseries editors: Daniel Alpay (Beer Sheva, Israel) Birgit Jacob (Wuppertal, Germany) Andre C.M. Ran (Amsterdam, The Netherlands)

Subseries Advances in Partial Differential Equations Subseries editors: Bert-Wolfgang Schulze (Potsdam, Germany) Michael Demuth (Clausthal, Germany) Jerome A. Goldstein (Memphis, TN, USA) Nobuyuki Tose (Yokohama, Japan) lngo Witt (Gottingen, Germany)

Honorary and Advisory Editorial Board: Lewis A. Coburn (Buffalo, NY, USA) Ciprian Foias (College Station, TX, USA) J. William Helton (San Diego, CA, USA) Thomas Kailath (Stanford, CA, USA) Peter Lancaster (Calgary, AB, Canada) Peter D. Lax (New York, NY, USA) Donald Sarason (Berkeley, CA, USA) Bernd Silbermann (Chemnitz, Germany) Harold Wid om (Santa Cruz, CA, USA)

Stability of Operators and Operator Semigroups Tanja Eisner

Birkhauser

Author: Tanja Eisner Arbeitsbereich Funktionalanalysis Mathematisches lnstitut Auf der Morgenstelle 10 72076 TO bingen Germany e-mail: [email protected]

2010 Mathematics Subject Classification: 47 A10, 47D06, 47 A35, 28D05 Library of Congress Control Number: 2010930173

Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de ISBN 978-3-0346-0194-8 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2010 Springer Basel AG P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCFoo

ISBN 978-3-0346-0194-8 987654321

e-ISBN 978-3-0346-0195-5 www.birkhauser-science.ch

Contents Introduction

1

I

5 5 5 7 8

Functional analytic tools 1 Compact semigroups . . . . . . . . . . . . 1.1 Preliminaries . . . . . . . . . . . . 1.2 Relatively compact sets in .Co-(X) . 1.3 Compact semitopological semigroups . 1.4 Abstract Jacobs-Glicksberg-de Leeuw decomposition for operator semigroups . . . . . . . . . . . . . . . . 1.5 Operators with relatively weakly compact orbits 1.6 Jacobs-Glicksberg-de Leeuw decomposition for Co-semigroups . . . . . 2 Mean ergodicity . . . . . . . . . . . . . . . 2.1 Mean ergodic operators . . . . . . 2.2 Uniformly mean ergodic operators 2.3 Mean ergodic Co-semigroups ... 2.4 Uniformly mean ergodic C0 -semigroups 3 Specific concepts from semigroup theory . . . . 3.1 Cogenerator . . . . . . . . . . . . . . . . 3.2 Inverse Laplace transform for C0-semigroups 4 Positivity in .C(H) . . . . . . . . . . . . . . . . 4.1 Preliminaries . . . . . . . . . . . . . . . 4.2 Spectral properties of positive operators 4.3 Implemented operators . . 4.4 Implemented semigroups .

II Stability of linear operators 1 Power boundedness . . . . . . . . . . 1.1 Preliminaries . . . . . . . . . 1.2 Characterisation via resolvent 1.3 Polynomial boundedness . . .

. . . .

9 12 12 13 13 18 20 22 23 23 26 32 32

33 34 35

37 37 37 40 43

Contents

vi 2

3

4

5

6

Strong stability . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . 2.1 Representation as shift operators 2.2 Spectral conditions . . . . . . . 2.3 Characterisation via resolvent . 2.4 Weak stability . . . . . . . . . . . . . Preliminaries . . . . . . . . . . 3.1 Contractions on Hilbert spaces 3.2 Characterisation via resolvent . 3.3 Almost weak stability . . . Characterisation .. 4.1 A concrete example 4.2 Category theorems . . . . Unitary operators 5.1 Isometries . . . . . 5.2 Contractions .. . 5.3 Stability via Lyapunov's equation . Uniform exponential and strong stability. 6.1 Power boundedness . . . . . . . . . . . . . 6.2

III Stability of C0 -semigroups Boundedness . . . . . . . . . . . . . . . . 1 Preliminaries . . . . . . . . . . . 1.1 Representation as shift semigroups 1.2 Characterisation via resolvent . 1.3 Polynomial boundedness . 1.4 Uniform exponential stability ... 2 Spectral characterisation .. 2.1 Spectral mapping theorems 2.2 Theorem of Datko-Pazy . 2.3 Theorem of Gearhart. 2.4 Strong stability . . . . . . . . . . 3 Preliminaries . . . . . . . 3.1 Representation as shift semigroups 3.2 Spectral conditions . . . . . . . 3.3 Characterisation via resolvent . 3.4 Weak stability . . . . . . . . . . . . . 4 Preliminaries . . . . . . . . . . 4.1 Contraction semigroups on Hilbert spaces 4.2 Characterisation via resolvent . 4.3 Almost weak stability . . 5 Characterisation 5.1 Concrete example 5.2

45

46 48

49 53 55 55

57 60 61 61

66 67 67 71

72

74 74 76

79 79 79 81 82 85 91 91 92 94 96 99 99 102 103 105 108 108 109 112 114 114 120

Contents

vii

6

122 123 126 127 129 129 130 132

7

Category theorems . . 6.1 Unitary case . 6.2 Isometric case . 6.3 Contractive case: remarks Stability via Lyapunov's equation . 7.1 Uniform exponential stability 7.2 Strong stability . 7.3 Boundedness . . . . . . . . .

IV Connections to ergodic and measure theory 1 Stability and the Rajchman property . . . . . . . . . . . . 1.1 From stability to the Rajchman property and back 1.2 Category result, discrete case .. 1.3 Category result, continuous case 2 Stability and mixing . . . . . . 2.1 Ergodicity . . . . . . . . 2.2 Strong and weak mixing 3 Rigidity phenomena . . . . . . 3.1 Preliminaries . . . . . . 3.2 Discrete case: powers of operators 3.3 Continuous case: Co-semigroups 3.4 Further remarks . . . . . . . . .

133 133 133 137 141 141 142 144 148 148 150

V Discrete vs. continuous 1 Embedding operators into C0 -semigroups . . . . . 1.1 Sufficient conditions via functional calculus 1.2 A necessary condition . . . . . . . . . . . . 1.3 Normal operators and projections on Hilbert spaces 1.4 Isometries and co-isometries on Hilbert spaces 1.5 Abstract examples: a residuality result . 2 Cogenerators . . . . . . . . . . . . . . . . . . . . . . . 2.1 (Power) boundedness . . . . . . . . . . . . . . . 2.2 Characterisation via cogenerators of the rescaled semigroups . . . . . . . . Examples . . . . . . . . . 2.3 Polynomial boundedness . 2.4 Strong stability . . . . . . 2.5 Weak and almost weak stability . 2.6

163 163 164

Bibliography

183

List of Symbols

201

Index

203

156 161

165 167 168 170 172

172 175 178 180 181 182

Introduction The real understanding involves, I believe, a synthesis of the discrete and continuous ... L. Lovasz, Discrete and Continuous: two sides of the same?

Evolving systems can be modelled using a discrete or a continuous time scale. The discrete model leads to a map 'P and its powers 'Pn on some state space n, while the continuous model is given by a (semi)fiow ('Ptk;,o on n satisfying 'Po = Id and 'Pt+s = 'Pt'Ps· In many situations, the state space n is a Banach space and the maps are linear and bounded. In this case, we will use the notations T and T(t) instead of 'P and 'Pt. In this book, our focus is on the various concepts of stability of such linear systems, where by stability we mean convergence to zero of {Tn}~=l or (T(t)k?.o in a sense to be specified. This property is crucial for study of the qualitative be• haviour of dynamical systems, even in the non-linear case. We adopt the general philosophy of a parallel treatment of both the discrete and the continuous case, systematically comparing methods and results. We try to give a reasonably com• plete picture mentioning (most of) the relevant results, a strategy that has, by the way, helped us to identify a number of natural open problems. However, we do not discuss the case of positive operators and semigroups on Banach lattices, referring the reader to, e.g., Nagel (ed.) [196] or the recent monograph of Emel'yanov [75] for this topic. We also tried to minimise overlap with the monographs of van Neer• ven [204] on asymptotics in the continuous case and Muller [191] in the discrete case. Instead we emphasise the connections of stability in operator theory to its analogues in ergodic theory and harmonic analysis. In the following we summarise the content of the book. Chapter I gives an overview on some functional analytic tools needed later. Besides the classical decomposition theorems of Jacobs~Glicksberg~de Leeuw for compact semigroups, we recall the powerful notion of the cogenerator of a Co• semigroup. Finally, we present one of our main concepts for the investigation of stability of C0 -semigroups, the Laplace inversion formula, which can also be seen as an extension of the Dunford functional calculus for exponential functions.

Introduction

2

In Chapter II we investigate the behaviour of the powers of a bounded linear operator on a Banach space. As a first step, we study power boundedness. Here and later, behaviour of the resolvent of the operator near the unit circle plays a crucial role. In particular, we give a resolvent characterisation for power boundedness on Hilbert spaces using the £ 2 -norm of the resolvent on circles with radius greater than 1. We further study the related notion of polynomial boundedness which is, surprisingly, easier to characterise than power boundedness. Stability is the topic in the rest of the chapter. While uniform stability is characterised by the spectral radius, we discuss strong stability and give various characterisations of it, both classical and recent. When turning to weak stability, we encounter a phenomenon, well-known in ergodic theory and described by Katok and Hasselblatt [143, p. 748] as follows 1

" ... It [mixing] is, however, one of those notions, that is easy and natural to define but very difficult to study... ". Thus very few results concerning weak stability are known, a sufficient condition in terms of the resolvent being one of the exceptions. We then introduce the concept of almost weak stability and give various equivalent conditions. From this we see that almost weak stability, while looking artificial, is much easier to characterise than weak stability. In particular, if the operator has relatively compact orbits, then almost weak stability is equivalent to "no point spectrum on the unit circle". Almost weak stability turns out to be fundamentally different from weak stability, as we see in Section 5. We show that a "typical" (in the sense of Baire) unitary operator, a "typical" isometry, and a "typical" contraction on a separable Hilbert space are almost weakly, but not weakly, stable. This is an operator the• oretic counterpart to the classical theorems of Halmos [116] and Rohlin [221] on weakly and strongly mixing transformations on a probability space. We close the chapter by characterising stability and power boundedness of operators on Hilbert spaces via Lyapunov equations. In Chapter III we turn our attention to the time continuous case and consider Co-semigroups (T(t) )t>O on Banach spaces. As in the previous chapter we discuss boundedness and stability of (T(t) )t>o and try to characterise these properties by the generator, its spectrum and resolvent. We start with boundedness and discuss in particular the quite recent characterisation of a bounded Co-semigroup on a Hilbert space by Gomilko [104] and Shi, Feng [231]. Their integrability condition for the resolvent on vertical lines (Theorem 1.11) is the key to the resolvent ap• proach to boundedness and stability, as first introduced by van Casteren [45, 46]. We then characterise generators of polynomially bounded semigroups (i.e., semi• groups growing not faster than a polynomial) in terms of various resolvent condi• tions. 1 Mixing

corresponds to weak stability.

Introduction

3

A discussion of uniform exponential stability of C0-semigroups follows, a notion which is more difficult to characterise than its discrete analogue. Besides Gearhart's generalisation to Hilbert spaces of Liapunov's stability theorem and the Datko-Pazy theorem, we extend Gearhart's theorem to Banach spaces. Strong stability is the subject of the following section containing the classical results of Sz.-Nagy and Foia§, Lax and Phillips and the more recent theorem by Arendt, Batty [9], Lyubich and Vu [180]. We then discuss the resolvent approach developed by Tomilov [243]. Weak stability and its characterisations, partly classical and partly quite recent, is the next topic. We emphasise that, as in the discrete case, weak stability is much less understood than its strong and uniform analogues, still leaving many open questions. For example, it is not clear how to characterise weak stability in terms of the resolvent of the generator. Next, we look at almost weak stability (Definition 5.3), which is closely re• lated to weak stability but occurs much more frequently. We give various equivalent conditions, analogous to the discrete case, and present a concrete example of an almost weakly, but not weakly, stable semigroup. We finally present category theo• rems analogous to the discrete ones stating that a "typical" (in the sense of Baire) unitary C0 -group as well as a "typical" isometric C0 -semigroup on a separable infinite-dimensional Hilbert space are almost weakly, but not weakly, stable. Characterisation of stability and boundedness of C0 -semigroups on Hilbert spaces via Lyapunov equations is the subject of the last section. Chapter IV relates our stability concepts to weakly and strongly mixing transformations and flows in ergodic theory and (via the spectral theorem) to Raj• chman and non-Rajchman measures in harmonic analysis. In addition, "typical" behaviour of contractive operators and semigroups on Hilbert spaces is studied using the notion of rigidity, extending the corresponding results in the previous two chapters. In Chapter V we build bridges between discrete systems {Tn}~=O and contin• uous systems (T(t))t~O· We start by embedding a discrete system into a continu• ous one. More precisely, we ask for which operators T there exists a Co-semigroup (T(t))t~o with T = T(l). We discuss some spectral conditions for embeddability and give classes of examples for which such a Co-semigroup does or does not exist. The general embedding question is still open as well as its analogues in ergodic and in measure theory. In Section 2 we discuss the connection between a Co-semigroup T(·) and its cogenerator V and its induced discrete system {Vn}~=O· On Hilbert spaces there are great similarities between asymptotic properties of the two sys• tems. We give classical and more recent results, discuss difficulties, examples, and open questions.

Acknowledgment. My deepest thanks go to Rainer Nagel, whose strong encour• agement and support, deep mathematical insight and permanent optimism has been an invaluable help.

4

Introduction

I am very grateful to Wolfgang Arendt, Ralph Chill, Balint Farkas, Jerome Goldstein, Sen-Zhong Huang, David Kunszenti-Kovacs, Yuri Latushkin, Mariusz Lemariczyk, Jan van Neerven, Werner Ricker, Yuri Tomilov, Ulf Schlotterbeck, Andras Sereny, Jaroslav Zemanek and Hans Zwart for helpful and interesting discussions and comments on this book. The cordial working atmosphere in the AGFA group (Arbeitsgemeinschaft Funktionalanalysis) in Ti.ibingen has been very stimulating and helpful. Special thanks go to Andras Batkai, Fatih Bayazit, Ulrich Groh, Retha Heymann, Bernd Kloss, David Kunszenti-Kovacs, Felix Pogorzelski, Ulf Schlotterbeck, Marco Schreiber, and Pavel Zorin for reading preliminary versions of the manuscript. Support by the Dr. Meyer-Struckmann Stiftung (via the Studienstiftung des deutschen Volkes), the European Social Fund and the Ministry Of Science, Re• search and the Arts Baden-Wi.irttemberg is gratefully acknowledged. I deeply thank my parents for their hearty encouragement and permanent support, as well as Anna M. Vishnyakova for teaching me how to become a math• ematician.

Chapter I

Functional analytic tools In this chapter we introduce some functional analytic tools needed later. We start with the theory of general compact semigroups and the decomposi• tion theorem of Jacobs-Glicksberg-de Leeuw for (discrete or continuous) operator semigroups. If applicable it states that an operator semigroup can be decomposed into a "reversible" and a "stable" part. Since the reversible part corresponds to an action of a compact group and can be described using harmonic analysis, it remains to study the stable part, the main goal of this book. We then discuss mean ergodicity of operators and semigroups and some con• cepts from semigroup theory such as the cogenerator and the inverse Laplace transform for C0 -semigroups.

1

Compact semigroups

1.1 Preliminaries We first recall some facts concerning the weak topology and weak compactness in Banach spaces and refer to Dunford, Schwartz [63, Sections V.4-6], Rudin [225, Chapter 3] and Schaefer [226, Section IV.ll] for details. We begin with the Eberlein-Smulian theorem characterising weak compact• ness in Banach spaces.

Theorem 1.1 (Eberlein-Smulian). For subsets of a Banach space, weak compact• ness and weak sequential compactness coincide. Next, we recall the following consequence of the Banach-Alaoglu theorem.

Theorem 1.2. A Banach space is reflexive if and only if its closed unit ball is weakly compact. In particular, every bounded set of a Banach space is relatively weakly com• pact if and only if the space is reflexive.

Chapter I. Functional analytic tools

6

Another important property of weakly compact sets is expressed by the Krein-Smulian theorem. Theorem 1.3 {Krein-Smulian). The closed convex hull of a weakly compact subset

of a Banach space is weakly compact. The following theorem characterises metrisability of the weak topology. Theorem 1.4. The weak topology on the closed unit ball of a Banach space X is

metrisable if and only if the dual space X' is separable. We recall that separability of X' implies separability of X, while the converse does not always hold. In addition, separable Banach spaces have a nice metrisabil• ity property with respect to the weak topology. Theorem 1.5. On a separable Banach space, the weak topology is metrisable on

weakly compact subsets. We now introduce the basic notations to be used in the following. For a linear operator T we denote by cr(T) , P(r(T), A(r(T) , R(r(T), r(T), p(T) and R(>-., T) its spectrum, point spectrum, approximative point spectrum and residual spectrum, spectral radius, resolvent set and resolvent at the point ).. E p(T), respectively. Here, the approximative point spectrum and residual spectrum are defined as

A(r(T) := P(r(T) U {A E C: rg(AJ- T) is not closed in X}

= {>-.EC: 3 {xn}

eX such that

llxnll = 1 and

lim IITxn- AXnll

n-+oo

= o},

R(r(T) := {>-. E C: rg(AJ- T) is not dense in X}. For more on these concepts see e.g. Conway [52, §VII.6] and Engel, Nagel [78, pp. 241-243]. For a C0 -semigroup (T(t))t:=::o we often write T(·), and denote its growth bound by w0 (T). The spectral bound s(A) of an (in general unbounded) operator A is s(A) := sup{Re >-. : ).. E cr(A)} and its pseudo-spectral bound (also called abscissa of the uniform boundedness of the resolvent) is

s0 (A) := inf{a E ~: R(A,A) is bounded on {A: Re>-. >a}}. Recall that for a C0 -semigroup T(-) with generator A the relations

s(A)::::; so(A) ::::; wo(T) hold, while both inequalities can be strict, see e.g. Engel, Nagel [78, Example IV.3.4] and van Neerven [204, Example 4.2.9], respectively. For other growth bounds and their properties see van Neerven [204, Sections 1.2, 4.1, 4.2].

1. Compact semigroups

7

1.2 Relatively compact sets in .Cu(X) In this subsection we characterise relatively weakly compact sets of operators and give some important examples. To this purpose we denote by £,.(X) the space of all bounded linear operators on a Banach space X endowed with the weak operator topology. The following characterisation goes back to Grothendieck. Lemma 1.6. For a set of operators T C £(X), X a Banach space, the following

assertions are equivalent.

(a) T is relatively compact in £,.(X). (b) T x := { Tx : T E T} is relatively weakly compact in X for all x E X.

(c) T is bounded, and T x is relatively weakly compact in X for all x in some dense subset of X.

Proof. We follow Engel, Nagel [78, pp. 512-514]. The implication (a)::::}(b) follows directly from the continuity of the mapping T ~ Tx for every x E X. The converse implication follows from the inclusion

(T, £,.(X)) c

IT (Tx"",

a)

xEX

and Tychonoff's theorem. The uniform boundedness principle yields the implication (b)::::}(c). (c)::::}(b): Take x EX and {xn}~=l CD converging to x, where D denotes the dense subset of X from (c). By the Eberlein-Smulian theorem (Theorem 1.1), it is enough to show that every sequence {Tnx }~=l C T x has a weakly convergent subsequence. Take a sequence {Tnx}~=l C Tx. For x1 ED, there exists a subsequence {nk,I}k"= 1 and a vector Z1 EX such that weak-limk--->oo Tnk,l x1 = z1. Analogously, for x2 E D there exists a subsequence {nk,2}~ 1 of {nk, 1}k"= 1 and a vector z2 E X such that weak-limk__.oo Tnk, 2 X2 = z2, and so on. By the standard diagonal procedure there exists a subsequence which we denote by {nk}~ 1 such that weak• limk--->oo TnkXm = Zm for every mEN. We have llzn- Zmll ~sup IITnk(Xn- Xm)ll ~ Cllxn- Xmll kEN

for C = sup{IITII: T E T} and all n,m EN. So {zn}~=l is a Cauchy sequence and therefore converges to some z E X. By the usual 3.s-argument we obtain 0 weak-limk__.oo TnkX = z. We now give some examples of relatively weakly compact subsets of operators.

Chapter I. Functional analytic tools

8

Example 1. 7. (a) On a reflexive Banach space X, any norm bounded family T relatively weakly compact by the Banach-Alaoglu theorem.

~

.C(X) is

(b) Let T ~ .C(L 1 (JL)) be a norm bounded subset of positive operators on the Banach lattice L 1 (JL), and suppose that Tu :S u for some JL-almost everywhere positive u E L 1 (J.L) and every T E T. Then T is relatively weakly compact since the order interval [-u, u] is weakly compact, T-invariant and generates a dense subset (see Schaefer [227, Thm. II.5.10 (f) and Prop. II.8.3]). (c) Let S be a semitopological semigroup, i.e., a (multiplicative) semigroup S which is a topological space such that the multiplication is separately con• tinuous (see Definition 1.8 below). Consider the space C(S) of bounded, continuous (real- or complex-valued) functions over S. For s E S define the corresponding rotation operator (Lsf)(t) := f(s · t). A function f E C(S) is said to be weakly almost periodic if the set {Lsf : s E S} is relatively weakly compact in C(S), see Berglund, Junghenn, Milnes [33, Def. 4.2.1]. The set of weakly almost periodic functions is denoted by W AP(S). If S is a compact semitopological semigroup, then C(S) = W AP(S) holds, see [33, Cor. 4.2.9]. This means that for a compact semitopological semigroup S the set {Ls : s E S} is always relatively weakly compact in .C(C(S)). We come back to this example in the proof of Theorem II.4.1, in Example II.4.10 as well as in the proof of Theorem III.5.1 and in Example III.5.9.

1.3 Compact semitopological semigroups In this subsection we present a very general and flexible approach to the quali• tative theory of operators and Co-semigroups. We will use the theory of compact semigroups and refer to Engel, Nagel [78, Section V.2] and Krengel [154, Section 2.2.4]. We call a set S with an associative multiplication · an abstract semigroup (S, ·).If the semigroup operation is fixed, we write only Sand st instead of (S, ·) and s · t.

Definition 1.8. An abstract semigroup S is called a semitopological semigroup if Sis a topological space such that the multiplication is separately continuous, i.e., such that the maps s f--7 stands f--7 ts are continuous for every t E S. If in addition S is compact, we call S a compact semigroup. A subset J C S is called an ideal if J S c J and S J C J hold. The following easy lemma shows that separate continuity of the multiplica• tion is enough to preserve the semigroup property and commutativity under the closure operation.

Lemma 1.9. Let S be a semitopological semigroup and T C S. If T is a subsemi• group, then so is T. Moreover, if T is commutative, then so is T.

1. Compact semigroups

9

We now state the classical theorem describing the structure of commutative compact semigroups. Theorem 1.10 (Structure of compact semigroups). LetS be a compact commuta•

tive semitopological semigroup. Then there exists a unique minimal ideal /( in S which is a compact topological group (i.e., the multiplication and the inverse maps are continuous from /C x /C ~ /C and /C ~ IC, respectively) and satisfies /( = qS for the unit element q of IC. Proof Observe first that, for closed ideals J1, ... , Jn inS, nj= 1 Jj is a nonempty closed ideal containing Jd2 · · · Jn. By compactness of S we conclude that IC:=

n

J

J closed ideal

is a nonempty closed ideal inS. We now show that /( is the minimal ideal. Let J be an arbitrary ideal in S and s E J. We observe that Js := sS is an ideal as well and is closed as the image of the compact setS under the multiplication by s. Hence, /( c Js c J and /Cis a minimal ideal. Uniqueness follows by the above representation of /C. We now check that /C is a group. Take s E /C. Then siC C /( is an ideal as well, hence

(I.l) by the minimality of /C. Therefore there exists q E /( with sq = s. We show that q is the unit in /C. Indeed, for an arbitrary r E /( there exists by (I.l) some r' E /( with sr' = r. So by commutativity of S we have rq = sr' q = r' s = r which implies that q is the unit element in /C. Existence of the inverse follows again from (I.l). The equality /( = qS follows from minimality of /( and the ideal property of

qS. We now invoke the Ellis theorem (see, e.g., Namioka [201] or Glasner [96, pp. 33-37]) stating that each compact semitopological group is in fact a topological 0 ~oup.

1.4 Abstract Jacobs-Glicksberg-de Leeuw decomposition for operator semigroups We now consider the case where Sis a subsemigroup of .C(X) (X a Banach space) considered with the usual multiplication. The typical examples for topologies mak• ing S a semitopological semigroup are the weak, strong and norm operator topolo• gies. If S is commutative and compact with respect to one of these topologies, the idempotent element q from Theorem 1.10 is a projection, hence

X= kerq EB rgq

10

Chapter I. Functional analytic tools

and both subspaces areS-invariant. Since the unit q of the minimal ideal K = qS is zero on ker q and the identity operator on rg q, we have

K = {0} EB Slrgq· Therefore, the semigroup S satisfies 0 E Slker q and Slrg q is a compact topological group. These properties describe the structure of commutative compact semigroups of operators on a Banach space. We state it for the weak operator topology only. Here and below, r denotes the unit circle in C. Theorem 1.11 {Abstract Jacobs-Glicksberg-de Leeuw decomposition for operator semigroups). Let X be a Banach space and let T c £{X) be a commutative, relatively weakly compact semigroup. Then X= Xr EB X 8 , where

Xr :=lin{ x EX : for every T E T there exists"( E r such that Tx Xs

:={ x

="(X}

EX: 0 is a weak accumulation point ofTx }.

Proof. Denote by S the closure of T in the weak operator topology. Then S is a compact commutative semitopological semigroup, see Lemma 1.9. Denote by K its minimal ideal and by q the unit inK. We show that Xs = kerq and Xr = rgq. Part 1: Xs = kerq.

We observe first that, by continuity of the mapping T ~---+ Tx, £ 17 (X) --. (X, u), we have Tx" = Sx for every x EX. To show kerq C Xs take x with qx = 0. Then 0 E Sx = Txa and hence x E X 8 • For the converse inclusion take x E X 8 • Then 0 E Txa = Sx and therefore there exists an operator RES with Rx = 0 and hence qRx = 0 as well. Since qR E K and K is a group, there exists an operator R' E K with R' qR = q. This implies qx = R' qRx = 0. Part 2: Xr

= rgq.

We first prove "c". Take X E X with T X c r . x. Then Sx c r . X as well and hence qx ="(X for some"( E f. (Indeed"(= 1 by q2 = q.) Therefore, x E rgq. By density of such vectors in Xr and closedness of rg q we have Xr C rg q. The proof of rg q C Xr uses some basic facts on compact abelian groups, see e.g. Hewitt, Ross [127]. Denote by m the Haar measure on K, i.e., the unique invariant probability measure with respect to multiplication. (For an elegant functional-analytic proof of its existence for compact groups see Izzo [133].) For a continuous character, i.e., a multiplicative continuous function X : K --. r, define Pxx :=

l

x(S)Sx dm(S)

1. Compact semigroups

11

weakly. (Note that the function S ~--+ x(S)(Sx, y) is continuous.) We obtain an operator Px : X ---+ X" and show that actually Px : X ---+ X. Indeed, for a fixed x E X the set M := {x(S)Sx, S E K} is weakly compact in X. Since the closed convex hull of a weakly compact set is again weakly compact by the Kreln-Smulian theorem (see Theorem 1.3), Pxx belongs to the image of co{M} in X" under the canonical embedding and hence represents an element of X. So Px E .C(X) with IIPx I ::; supSEK IISII· We now show that rg(Px) c {x: Tx c fx}. TakeR E K. By the invariance of m and multiplicativity of x,

RPxx = =

l

x(S)RSx dm(S) = x(R)

x(R)

l

l

x(RS)RSx dm(S),

x(S)Sx dm(S) = x(R)Pxx

holds for every x E X, i.e., RPx = x(R)Px· For R := q this means qPx = x(q)Px = Px, hence rg(Px) C rgq. Take now SET and defineR:= Sq. Then we have SPx = SqPx = x(Sq)Px and therefore

rg(Px)

C

{x: Sx = x(Sq)x for every SET}

(1.2)

and in particular rg(Px) C Xr. It remains to show that

To this purpose, take y E X' vanishing on rg Px for every character x and show that y vanishes on rgq. By assumption JK x(S)(Sx, y)dm(S) = 0 for every charac• ter x and x E X. By the continuity of the integrand and by the fact that the linear hull of all characters is dense in L 2 (K, m) (see e.g. Hewitt, Ross [127, Theorem 22.17]), we obtain (Sx, y) = 0 for every S E K. In particular, (qx, y) = 0 for every x, soy vanishes on rgq and the theorem is proved. D The subspaces Xr and Xs in the above decomposition are called reversible and stable subspaces, respectively.

Remark 1.12. A function ..\ : T ---+ C is called an eigenvalue function if there exists 0 # x E X such that Tx = ..\(T)x holds for every T E T. By (1.2) above, we see that every character x with Px # 0 defines an eigenvalue function ..\ : K ---+ r by ..\(T) = x(Tq) with eigenvectors in rgPx. Conversely, if for some x # 0 the equalities Tx = ..\(T)x with ..\(T) E r hold for every T E K, then TSx = ..\(S)Tx = ..\(S)..\(T)x and T ~--+ ..\(T) is continuous. Thus every such eigenvalue function ..\ is a character on K.

Chapter I. Functional analytic tools

12

1.5

Operators with relatively weakly compact orbits

We now apply the abstract setting to discrete operator semigroups.

Definition 1.13. An operator T on a Banach space X has relatively weakly compact orbits if the set T := {Tn : n E N0 } satisfies one of the equivalent conditions in Lemma 1.6. Example 1.14. By the Banach-Alaoglu theorem, every power bounded operator on a reflexive Banach space has relatively weakly compact orbits. We now apply the abstract decomposition from Subsection 1.4 to such oper• ators.

Theorem 1.15 (Jacobs-Glicksberg-de Leeuw decomposition). Let X be a Banach space and letT E .C(X) have relatively weakly compact orbits. Then X= Xr E£)X8 , where Xr:=lin{x EX: Tx="fxforsome"(Ef},

Xs

:={ x EX: 0 is

a weak accumulation point of {Tnx: n

= 0, 1, 2, ... } }.

We will formulate an extended version of this theorem later in Theorem 11.4.8. Finally, we state the Jacobs-Glicksberg-de Leeuw decomposition theorem for the strong operator topology.

Theorem 1.16. Let X be a Banach space and T E .C(X) such that for every x E X the orbit {Tnx: n = 0, 1, 2 ... } is relatively compact in X. Then X= Xr EElXs for

Xr := lin{x EX: Tx = "fX for some 'Y E r}, Xs := {x EX: IITnxll---t 0 as n ---too}. The prooffollows directly from Theorem 1.15, the fact that on compact sets weak convergence implies convergence, and since limj-4oo IITni xll = 0 for some subsequence {nj} implies limn--+oo IITnxll = 0, see Lemma II.2.4.

1.6 Jacobs-Glicksberg-de Leeuw decomposition for C0-semigroups We now apply the abstract setting of Subsection 1.4 to C0 -semigroups. Note that the results of this section are completely analogous to the discrete version consid• ered in Subsection 1.5.

Definition 1.17. A C0 -semigroup T(·) on a Banach space X is called relatively weakly compact if the set T := {T(t): t 2: 0} c .C(X) satisfies one of the equivalent conditions in Lemma 1.6. Remark 1.18. A C0-semigroup T(·) on a Banach space X is relatively weakly compact if and only if T(l) has relatively weakly compact orbits. The non-trivial "if" implication follows from the semigroup law and the fact that the sets {T(r)x: r E [0, 1]} are (strongly) compact as the continuous image of [0, 1].

2. Mean ergodicity

13

The following theorem is again a special case of the abstract Jacobs-Clicks• berg-de Leeuw decomposition and is fundamental for the stability theory of C 0 semigroups, see Engel, Nagel [78], Theorem V.2.8. Theorem 1.19 (Jacobs-Glicksberg-de Leeuw decomposition for C0 -semigroups). Let X be a Banach space and T(·) be a relatively weakly compact Co-semigroup on X. Then X= Xr EB X 8 , where

Xr

:= lin{x EX:

Xs

:= { x EX : 0

T(t)x

=

eicdx for some a E ffi. and all t 2:: 0},

is a weak accumulation point of {T(t)x: t 2:: 0} }.

In Theorem III.5. 7 we formulate an extended version of the above theorem. Moreover, we refer to Arendt, Batty, Hieber, Neubrander [10, Theorem 5.4.11] for an individual version. We now state the decomposition theorem of Jacobs-Glicksberg-de Leeuw for C0-semigroups with respect to the strong operator topology. Theorem 1.20. Let X be a Banach space and T(·) be relatively compact in the stmng operator topology, i.e., for every x EX the orbit {T(t)x: t 2:: 0} is relatively compact in X. Then X= Xr EB Xs for

Xr := lin{x EX: T(t)x = eiatx for some a E ffi. and all t 2:: 0}, Xs := {x EX: IIT(t)xll--t 0 as t ___, oo}. The prooffollows as in the discrete case directly from Theorem 1.19, the fact that, for sequences in a compact set, weak convergence implies convergence, and Lemma III.3.4. For an individual version of Theorem 1.20 see Arendt, Batty, Hieber, Neu• brander [10, Theorem 5.4.6]. Remark 1.21. Theorem 1.20 applies, for instance, to eventually compact C 0 semigroups or bounded Co-semigroups whose generator has compact resolvent, see Engel, Nagel [78, Corollary V.2.15].

2 Mean ergodicity We now introduce mean ergodicity, or convergence in mean, of operators and C0-semigroups. This is an important asymptotic property having its origin in the classical "mean" and "individual ergodic theorems" by von Neumann and Birkhoff (see books on ergodic theory such as Halmos [118], Petersen [212], Walters [254]) which occurs in quite general situations.

2.1 Mean ergodic operators We first look at mean ergodicity of operators and follow Yosida [260, Section VIII.3], Nagel [195] as well as Engel, Nagel [78, Section V.4] where the continuous case is treated.

Chapter I. Functional analytic tools

14

We begin with the so-called Cesaro means of an operator.

Definition 2.1. For a bounded operator T on a Banach space X the Cesaro means Sn are defined by

We are interested in convergence of Sn. The following easy lemma already gives some information. Lemma 2.2. Let T be a bounded operator on a Banach space X. If T satisfies

limn-+oo

IITnxll n = 0 for every x E X, then Snx converges as n

----t

oo for every

x E FixT E9 rg(J- T).

More precisely, Snx = x for every x E Fix T and limn-+oo Snx = 0 for every T).

x E rg(I-

Proof. To prove the second statement take x = z- Tz E rg(I- T). Then we have

0

by assumption.

Definition 2.3. A bounded operator T on a Banach space X is called mean ergodic if the Cesaro means {Snx} ~=o converge as n ----t oo for every x E X. In this case the limit

x

f-t

Px := lim Snx n-+oo

is called the mean ergodic projection corresponding to T. 1) For a mean ergodic operator T, the operator P is indeed a projection commuting with T since P = T P = PT follows from the definition and implies P = Tn P = liffin_, 00 n~l I:~=O Tk P = P 2. If T is a contraction on a Hilbert space, then Pis an orthogonal projection since it is contractive.

Remark 2.4.

2) By T:x = Snx- n;:;-l Sn_ 1 x, every mean ergodic operator T satisfies lim IITnxll = 0 n-+oo n

for every x EX,

and hence r(T) ::::; 1 by the uniform boundedness principle. As a counterpart to the Cesaro means we now introduce Abel means of an operator.

2. Mean ergodicity

15

Definition 2.5. For a bounded operator T on a Banach space X, the Abel means of T are the operators Br defined by -

Srx := (r- 1) for r

L

00

n=O

rnx rn+l,

x EX

> 1, whenever the above series converges.

The following classical property is central for our study, see, e.g., Hardy [123, Section V.12], Emilion [76] or Shaw [230] for the continuous case. Lemma 2.6 (Equivalence of the Cesaro and Abel means). Let {an}~=l be a seoo

every r > 1. Then convergence of the Cesaro means as n --t of the Abel means as r --t 1+ and the limits coincide, i.e., 1

n

L

r~: 1 exist for n=O oo implies convergence

quence in a Banach space X such that the Abel means (r- 1)

oo

lim - - '"'ak = lim (r- 1)'"' a+nl. ~ rn r--+1+ n-+oo n + 1 ~ n=O k=O

Conversely, convergence of the Abel means implies convergence of the Cesaro means and the limits coincide in each of the following cases:

• {an}~==l is bounded; • X = C and an 2 0 for every n E N. In particular, for a bounded operator T on a Banach space with r(T) ::; 1, one has n 1 lim - - '"'Tk = lim (r- 1)R(r, T) r--+1+ n-+oo n + 1 ~ k=O

in the weak, strong and norm operator topology, whenever the left limit exists. Note that the last assertion follows from the previous ones by the Neumann series representation of the resolvent.

Remark 2.7. Shaw [230] showed that convergence of Abel means implies conver• gence of the Cesaro means and the limits coincide for every positive sequence in a Banach lattice. We now state an easy but useful property of mean ergodic operators. We will see in Theorem 2.9 that it is in fact equivalent to mean ergodicity. Proposition 2.8. Let T be a mean ergodic operator on a Banach space X. Then

the mean ergodic projection P yields a decomposition X

= rg P

EB ker P

Chapter I. Functional analytic tools

16 with

rgP = FixT,

ker P = rg(I- T).

Moreover, the projection P can also be obtained as Px = lim (r- 1)R(r, T)x

+

r--> 1

for all x EX.

(1.3)

Proof. The inclusion Fix T C rg P follows from the definition of P. For the con• verse inclusion take some z = Px E rgP. Then Tz = TPx = Px = z by the above remark. The inclusion rg(I- T) C ker P follows from Lemma 2.2. For the converse inclusion take y E X' vanishing on rg(J- T). This means that (x, y) = (Tx, y) for every x EX, which implies (x,y) = (Px,y) for every x EX, hence y vanishes on ker P, and we obtain ker PC rg(I- T). The last formula follows from r(T) 1 and Lemma 2.6. 0

s

The following classical theorem gives different characterisations of mean er• godicity, see, e.g., Yosida [260, Theorem VIII.3.2], Nagel [195] and Krengel [154, § 2.1]. Theorem 2.9 (Mean ergodic theorem). LetT be a bounded operator on a Banach space X satisfying supnEN IITnll < oo. Then the following assertions are equivalent.

(i) T is mean ergodic. (ii) For every x EX there exists a subsequence {nk}k=l such that SnkX converges weakly as k ____, oo. (iii) limr_,H(r -1)R(r,T)x exists for every x EX. (iv) FixT separates Fix(T'). (v) X= FixT EB rg(J- T). In particular, operators with relatively weakly compact orbits are mean ergodic. Proof. The equivalence (i){:}(iii) follows from Lemma 2.6. The implication (i):::}(ii) is trivial. For the implication (ii):::}(iv) take 0 i- y E FixT'. We have to find x E FixT with (x, y) # 0. Take x 0 E X with (x 0 , y) # 0. By (ii) there exists weak• limk_,oo Snkxo =: x for some subsequence {nk}k'=o· We now show that x E FixT. Observe

Since Tis power bounded, the second summand on the right-hand side tends to 0 as k ____, oo. Thus weak continuity of (I-T) implies x- Tx =weak- lim (I- T)(x- Snkxo) = 0, k-->oo

2. Mean ergodicity

17

hence x E Fix T. Therefore we obtain (x, y) = lim (SnkXo, y) = (xo, y) k-too

-=/-

0,

where we have used y E ker T', and (iv) follows. Assume now that (iv) holds. For (v) it suffices to show that the subspace G := FixT E8 rg(J- T)

is dense in X. Take y EX' vanishing on G. Then (x, y) = (Tx, y) for every x EX, and hence y E Fix T'. Since y vanishes on Fix T by assumption as well, we have by (iv) that y = 0, and density of G follows. To prove (v)=}(i) we first observe that Snx converges for every x E FixT E8 rg(J- T) by Lemma 2.2. By density of this set and boundedness of {Sn}~=O we obtain that Sn converges on the whole X, i.e., Tis mean ergodic. In order to show the last assertion, suppose that T has relatively weakly compact orbits and check assertion (ii). Take x E X and observe that {Snx : n E N} c co{Tnx : n = 0, 1, 2, ... } and hence is relatively weakly compact by assumption and the Krein-Smulian theorem. By the Eberlein-Smulian theorem, weak compactness coincides with weak sequential compactness, and assertion (ii) of Theorem 2.9 follows. D The following result shows the value of the various characterisations in the above mean ergodic theorem. Corollary 2.10. A power bounded operator T on a Banach space is mean ergodic

with mean ergodic pmjection P = 0 if and only if Fix T' = { 0}. The proof uses (i){:}(iv) in Theorem 2.9 and the fact that FixT' separates Fix T for power bounded operators. For a quite large class of operators, mean ergodicity holds automatically as a consequence of the Banach-Alaoglu theorem and Theorem 2.9. Corollary 2.11. Every power bounded operator on a reflexive Banach space is mean

ergodic. Note that mean ergodicity of power bounded operators characterises reflex• ivity for Banach spaces with unconditional basis, see Fonf, Lin, Wojtaszczyk [89] and also the discussion in Emel'yanov [75, Section 2.2.3].

n r C {1} and T has relatively weakly compact orbits, then the mean ergodic projection coincides with the projection from the Jacobs• Glicksberg-de Leeuw decomposition, see Section 1.5.

Remark 2.12. If P,:r(T)

There are many extensions of the mean ergodic theorem, see e.g. Berend, Lin, Rosenblatt, Tempelman [31] for recent results on modulated and subsequential ergodic theorems and further references.

Chapter I. Functional analytic tools

18

2.2 Uniformly mean ergodic operators In this subsection we consider a stronger property than the above mean ergodicity, see also Krengel [154, § 2.2]. Definition 2.13. An operator Ton a Banach space is called uniformly mean ergodic if the Cesaro means Sn converge in the norm operator topology.

In other words, an operator T is uniformly mean ergodic if and only if T is mean ergodic and one has lim IISn - Pll = 0 n-+oo

for the mean ergodic projection P. Remark 2.14. A uniformly mean ergodic operator T again satisfies r(T) ::; 1, and the projection P can be obtained as

P = lim (r- 1)R(r, T) r--+1+

by Lemma 2.6. Recall that for a mean ergodic operator this formula holds only pointwise. The following result characterises uniform mean ergodicity, see Lin [168]. Theorem 2.15. Let T be a power bounded operator on a Banach space. Then the following assertions are equivalent.

(i) T is uniformly mean ergodic. (ii) There exists limr-.H(r- 1)R(r, T). (iii) 1 E p(T) or 1 is a first-order pole of R(.>..,T). (iv) rg(I- T) is closed in X. (v) X= FixT EEl rg(I- T).

Proof. The equivalence (i){:}(ii) again follows from Lemma 2.6. (i)=>(iii). Assume that 1 E a(T). We have to show that 1 is a first-order pole of the resolvent. By Proposition 2.8, the decomposition into invariant subspaces X= FixT EEl rg(I- T) holds. Since TIFixT =I, and hence 1 is a first-order pole of R(>., TIFixT ), it suffices to show that 1 E p(Tiker P ).

(I.4)

Assume the opposite, i.e., that 1 E a (TikerP). Since Tis power bounded, we have 1 E Aa (TikerP)· Thus there exists an approximate eigenvalue {xm};;;'= 1 C ker P, llxmll = 1, such that limm-. 00 II(I- T)xmll = 0. Then we obtain n-1

lim (I- Tn)Xm

m--+oo

= m--+oo~ lim ""'Tk(I- T)xm = 0 k=O

2. Mean ergodicity

19

for every n E N. This implies limm_, 00 (I- Sn)Xm = 0, so 1 E Aa(Sn) and hence IISnlker PII ~ 1 for every n EN. This contradicts the uniform ergodicity ofT which means limn--+oo II Sn lker p II = 0. Hence (I.4) is proved. The implication (iii)=?(ii) is clear. (ii)=?(iv). Let P := limr_,H(r- 1)R(r, T). We first prove that Px = 0 for every x E rg(I- T). Take x = z- Tz E rg(I- T). Then we have, by the resolvent identity,

(r- 1)R(r, T)x = (r- 1)R(r, T)(I- T)z = (r- 1)z- (r- 1) 2 R(r, T)z and hence by (iii) Px = limr_,H(r -1)R(r, T)x = 0. SoP= 0 on Y := rg(I- T), i.e., lim II (r- 1)R(r, T)IYII = 0 r--+1+ holds. (We used here that Y is R(r, T)-invariant by the Neumann representation of the resolvent.) This implies that the operator

(I- T)R(r, T) =I- (r- 1)R(r, T) is invertible on Y for every r

> 1 small enough.

So we have in particular

Y = rg(I- T) = (I- T)R(r, T)Y c rg(I- T), and rg( I - T) is closed. (iv)=?(v). We first show that (I- T)lrg(J-T) is invertible. Assume the op• posite, i.e., that 1 E a(Tirg(J-r))· As in the proof of (i)=?(iii), this implies that 1 E a(Snlrg(J-T)) and hence IISnlrg(J-T)II ~ 1 for every n EN. On the other hand, for x = z- Tz E rg(I- T) we have, by

(n

+ 1)Snx =

n

L(Tkz- Tk+lz) = z- rn+lz k=O

and power boundedness ofT, that supnEN llnSnxll < oo. By closedness of rg(I- T) and the uniform boundedness principle this implies supnEN llnSnlrg(J-T}il < oo and hence limn--+oo IISnlrg(J-T)II = 0, a contradiction. So (I- T)lrg(J-T) is invertible. Take now x E X. By invertibility of (I- T)lrg(I-T) there exists a unique z E rg(I -T) such that (I -T)x =(I -T)z. Then (x-z) E FixT and x = (x-z)+z, and the decomposition X= FixT EB rg(I- T) is proved. (v)=?(i). Observe that FixT and rg(I- T) are invariant closed subspaces. On FixT the restriction of Sn is equal to I, so TIFixT is uniformly mean ergodic. Moreover, as in the proof of the implication (iv)=?(v) we have (by closedness of

rg(I- T)) lim IISnlrg(J-T)II = 0. n--+oo So T is uniformly mean ergodic by (v).

0

20

Chapter I. Functional analytic tools A direct corollary of the above characterisation is the following.

Corollary 2.16. A quasi-compact power bounded operator T on a Banach space is

uniformly mean ergodic.

Proof. Since T is quasi-compact, we have a decomposition into invariant subspaces X= X1 EBX2 and T = T1 EBT2 such that dimX1 < oo and r(T2 ) < 1. Assume that 1 E a-(T) which is equivalent to 1 E O"(T1 ). Since T1 is a power bounded matrix, we see by Jordan's representation that 1 is the first-order pole of R(>-., Tl). Thus T satisfies condition (iii) of Theorem 2.15. D Remark 2.17. For positive operators on Banach lattices a stronger assertion holds: A positive operator T is quasi-compact if and only if it is mean ergodic and satisfies

dim Fix T < oo, see Lin [170].

2.3

Mean ergodic C0-semigroups

We now characterise mean ergodicity of C0 -semigroups. Most of the proofs are analogous to the discrete case, so we omit them and refer to Engel, Nagel [78, Section V.4] for details. We first define Cesaro means of a C0 -semigroup T(·).

Definition 2.18. The Cesaro means of a Ca-semigroup T(·) are the operators

11t

S(t)x :=-

t

0

x EX and t > 0.

T(s)xds,

The following easy lemma describes convergence of the Cesaro means on a subspace. Here and later we use the notation FixT(·) :=

n

FixT(t).

t;:::o

Lemma 2.19. LetT(·) be a Ca-semigroup on a Banach space X. If T satisfies limt-too IIT(2xll = 0 for every x EX, then S(t)x converges as t-+ oo for every x E FixT(·) EBlin Ut;:::o rg(I- T(t)).

More precisely, S(t)x = x for every x E FixT(·) and limt-too S(t)x = 0 for every x E lin Ut>O rg(I- T(t)). Mean ergodic semigroups are defined as follows.

Definition 2.20. A C0-semigroup T(·) on a Banach space X is called mean ergodic if the Cesaro means S(t)x converge as t -+ oo for every x E X. In this case the limit

x

r-.

Px := lim S(t)x t-+oo

is called the mean ergodic projection corresponding to T (·).

21

2. Mean ergodicity

Remark 2.21. Mean ergodicity ofT(·) implies P = T(t)P = PT(t) and there• T(s)Pxds = P 2 x. So Pis indeed a projection fore Px = T(t)Px = limt-+oo T(·). with commuting

t J;

As above, we also consider the Abel means.

Definition 2.22. The Abel means of a Co-semigroup T(·) with w0 (T) operators Sa defined by

Sax:= a

loco e-asT(s)xds

~

0 are the

for x EX and a> 0.

The following relation between convergence of the Cesaro and Abel means is essential for the resolvent approach to asymptotics of C0 -semigroups, see, e.g., Emilion [76] and Shaw [230].

Lemma 2.23 (Equivalence of the Cesaro and Abel means, continuous case). Let X be a Banach space and f : ~+ --+ X be continuous such that the Abel means a 000 e-at f(t)dt exist for every a > 0. Then convergence of the Cesaro means as t --+ oo implies convergence of the Abel means as a --+ 0+ and the limits coincide, i.e.,

J

r

f(s)ds = lim a roo e-at f(t)dt. lim ~ a-+0+ } 0 t-+oo t }0 Conversely, convergence of the Abel means implies convergence of the Cesaro means and the limits coincide in each of the following cases: • f is bounded;

• X

= C and f (s) ~ 0 for

every s ~ 0.

In particular, for a Co-semigroup T(·) on a Banach space X with generator A satisfying wo(T) ~ 0 one has

~

r

T(s)xds = lim aR(a, A)x lim a-+0+ t-+oo t }0 for every x E X, whenever the left limit exists. The following proposition gives more information on the decomposition ob• tained from the mean ergodic projection.

Proposition 2.24. LetT(·) be a mean ergodic Co-semigroup on a Banach space X satisfying limt-+oo IIT(:)xll = 0 for every x E X. Then the mean ergodic projection P yields the decomposition X = rg P EB ker P

with

rgP = ker A= FixT(·),

ker P = rgA =lin Ut~o rg(J- T(t)).

22

Chapter I. Functional analytic tools

Moreover, the projection P can be obtained as Px = lim aR(a, A)x a--->0+

for all x EX.

(I.5)

The following classical theorem characterises mean ergodicity in various ways. Theorem 2.25 (Mean ergodic theorem for C0 -semigroups). LetT(·) be a bounded

Co -semigroup on a Banach space X. Then the following assertions are equivalent.

(i) T(·) is mean ergodic. (ii) For every x

E X there exists a sequence {tk}k=l that S(tk)x converges weakly as k -+ oo.

c IR+ converging to oo such

(iii) There exists lima--->O+aR(a,A)x for every x EX. (iv) ker A separates ker A'. (v) X= ker A EB rgA.

In particular, every relatively weakly compact C 0-semigroup, and hence every bounded semigroup on a reflexive Banach space, is mean ergodic. As a direct corollary we obtain an easy characterisation of C0 -semigroups whose Cesaro means converge to 0. Corollary 2.26. A bounded C0-semigroup T(·) on a Banach space with generator

A is mean ergodic with mean ergodic projection P

= 0 if and only if ker A' = {0}.

Remark 2.27. If T(·) is a relatively weakly compact semigroup with generator A

and Pa(A) ni!R = {0}, then the mean ergodic projection corresponding toT(·) co• incides with the projection from the Glicksberg-Jacobs-de Leeuw decomposition, see Section 1.6.

2.4 Uniformly mean ergodic C0-semigroups We now introduce uniformly mean ergodic C0 -semigroups. Again, the situation is similar to the discrete case and we refer to Engel, Nagel [78, Section V.4] for the proofs. Definition 2.28. A C0 -semigroup T(·) on a Banach space is called uniformly mean

ergodic if the Cesaro means S (t) converge as t

-+

oo in the norm operator topology.

Remark 2.29. A C0 -semigroup T(·) is uniformly mean ergodic if and only if it is

mean ergodic and lim IIS(t)- Pll = 0 t--->00 holds for the mean ergodic projection P. The following theorem characterises uniform mean ergodicity of C0-semi• groups, see Lin [169].

3.

Specific

concepts

from semigroup

23

theory

Theorem 2.30. LetT(·) be a bounded C0 -semigroup on a Banach space. Then the

following assertions are equivalent.

(i) T(·) is uniformly mean ergodic. (ii) There exists lima---+O+ aR(a, T). (iii) 0 E p(A) or 0 is a first-order pole of R(.\, A). (iv) rgA is closed in X. (v) X= FixT(·) EBrgA.

In this case, the mean ergodic projection satisfies P = lima---+O+ aR( a, T). A direct corollary of the above characterisation is the following example of uniformly mean ergodic semigroups. Corollary 2.31. A quasi-compact bounded Co-semigroup T(·) on a Banach space

is uniformly mean ergodic.

3 Specific concepts from semigroup theory In this section we present tools and methods from semigroup theory which we will need later. The main reference for basic definitions and facts from semigroup theory is Engel, Nagel [78].

3.1

Cogenerator

Another useful tool, besides the generator, in the theory of C0 -semigroups is the so• called cogenerator. It can be obtained easily from the generator (see formula (1.6) below), it is a bounded operator, and, as we will see later, reflects many properties of the semigroup. The diagram on top of the next page shows the major objects related to a Co-semigroup. The cogenerator is defined as follows. Definition 3.1. Let A generate a C0-semigroup T(·) on a Banach space X satisfying 1 E p(A). The operator V defined by

V

:=

(A+ I)(A- J)- 1

E

.C(X)

is called the cogenerator ofT(·). Remark 3.2. The identity

V =(A- I+ 2I)(A- J)- 1 =I- 2R(l, A)

(1.6)

implies that V-I has a densely defined inverse (V-I) - 1 = ~ (A- I). In particular,

A= (V + I)(V- J)- 1 =I+ 2(V- I)- 1 holds, i.e., the generator is also the (negative) Cayley transform of the cogenerator.

Chapter I. Functional analytic tools

24

(T(t))t;:=:o semigroup

v

(A,D(A))

cogenerator

generator

~/ (R(>.., A)h.Ep(A) generator's resolvent

Note that the cogenerator determines the generator, and therefore the semi• group, uniquely. As a further consequence of (I.6) one has the following easy description the spectrum and resolvent of V. Proposition 3.3. The spectrum of the cogenerator V is

>..- 1 : >..Ea(A) } . a(V)\{1}= { >..+1 The same relation holds for the point spectrum, residual point spectrum and ap• proximative point spectrum, respectively. Moreover, _ 11 , V ) R(>.., A) = >.. _1 1 (J- V)R ()..>.. +

(I.7)

holds for every>.. E p(A) \ {1}. Proof. The assertion on the spectrum of Vis a direct consequence of (I.6) and the spectral mapping theorem for the resolvent (see e.g. Engel, Nagel [78, Theorem IV.l.13]), while (!.7) follows from >.I- A= (>..V- >.I-V- I)(V- I)- 1

= (V(>..- 1)- (>.. + 1))(V- I)- 1 = (>..- 1) (>.. + l >..-1 for every >..

f:.

1.

v) (I- Vt

1

0

It is remarkable that for operators on Hilbert spaces there is a very simple characterisation of operators being the cogenerator of a contraction semigroup. This is the following theorem due to Foia§, Sz.-Nagy [238, Theorem III.8.1]. The

3.

Specific concepts

from semigroup

25

theory

proof below of the non-trivial "if" -implication (see also Katz [145]) is simpler than the original one which uses a special functional calculus to construct the semigroup.

Theorem 3.4 (Foia§-Sz.-Nagy). Let H be a Hilbert space and V E .C(H). Then V is the cogenerator of a contractive Co -semigroup if and only if V is contractive and 1 ~ Pa(V).

Proof. Assume first that Vis the cogenerator of a contractive Co-semigroup T(·) and denote by A its generator. Then the operator I- V = 2R(1, A) is injective and hence 1 ~ P11 (V). Moreover, by the Lumer-Phillips Theorem (see, e.g., Engel, Nagel [78, Theorem II.3.15]) A is dissipative and hence

I (A+ I)xll 2 - I (A- I)xll 2 = 4Re (Ax, x) : : ; 0 Therefore IIVxll = II(A + I)(Ax E H, hence Vis contractive.

Vx E D(A).

I)- 1 xll : : ; II(A- I)(A- n- 1 xll = llxll

for every

Let now V be a contraction with 1 ~ P11 (V). Then the operator I- V is injective, and we can define

A := -(I+ V)(I- V)- 1

with

D(A)

:= rg(I-

V).

Note that A= I - 2(I- V)- 1 holds. We show first that Re (Ax, x) ::::; 0 for every x E D(A). Indeed, for x E D(A) and y := (V- I)- 1 x we have

(Ax, x) = ((I+ V)(V- I)- 1 x, x) = ((I+ V)y, (V- I)y)

=

IIVYII 2 -IIYII 2 + 2i · Im (y, Vy)

and therefore Re (Ax, x) ::::; 0. We observe further that (I - A)- 1 = ~(I- V) and therefore 1 E p(A). Moreover, since Vis mean ergodic, we have rg(I- V) = H by Proposition 2.8, so the operator A is densely defined. The assertion now follows directly from the Lumer-Phillips Theorem (see, e.g., Engel, Nagel [78, Theorem II.3.15]). D Further, many properties of a contraction semigroup on a Hilbert space can be seen from its cogenerator. The following is again due to Foia§, Sz.-Nagy, see [238, Sections III.8-9].

Theorem 3.5. LetT(·) be a contractive Co-semigroup on a Hilbert space with co• generator V. Then T(·) is normal, self-adjoint, isometric or unitary if and only if V is normal, self-adjoint, isometric or unitary, respectively. Note that all the equivalences except the isometry property follow easily from the spectral theorem in its multiplicator form, see e.g. Conway [52, Theorem IX.4.6].

26

Chapter I. Functional analytic tools

The cogenerator approach allows us to transfer many properties of single operators to C0 -semigroups in a short and elegant way. However, it is not yet clear how this approach extends to Co-semigroups on general Banach spaces. To conclude this subsection we present an elementary but useful formula for the powers of the cogenerator in terms of the semigroup, see e.g. Gomilko [105].

Lemma 3.6. LetT(·) be a Co-semigroup on a Banach space X with wo(T) cogenerator V. Then

vnx

=X-

21

00

< 1 and (I.8)

L;._1 (2t)e-tT(t)xdt

holds for every x E X, where L; (t) is the first generalized Laguerre polynomial given by

~

L1(t)= n

L.....t

m=O

(-l)m ( n+1 )tm. m! n-m

(I.9)

Proof. By V =I- 2R(1, A) and the Laplace representation for the resolvent we obtain

Vnx =(I- 2R(1, A)tx = x- "n ' ( 2m )' ( n ) R(m- 1)(1, A)x L.....t m-1. n-m m=1

=X-

=

x-

~

L.....t m=1

21

2m( -1)m-1 ( n ) (m-1)! n-m 00

1oo e-ttm-1T(t)xdt 0

L;._ 1(2t)e-tT(t)xdt, D

and the lemma is proved.

3.2

Inverse Laplace transform for C0-semigroups

Our main tool in the resolvent approach to stability of Co-semigroups is a formula for the inverse Laplace transform of the semigroup. We first need a lemma on the behaviour of the resolvent of an operator on the left of its abscissa of uniform boundedness.

Lemma 3.7. Let A be a closed operator with so(A) IIR(z, A)xll

--t

0

as lzl

--t

< oo. Then

oo, Re z ~ a

(1.10)

holds for every a> so(A) and x EX. Proof. Take any a > so (A). Then there exists a constant M > 0 such that IIR(z,A)II ~ M for all z E C with Rez ~ a. Take now x E D(A) and z with Rez ~a. Then 1 1 IIR(z, A)xll = ~llx + R(z, A)Axll :::; ~(llxll

+ MIIAxll),

3. Specific concepts from semigroup theory

27

and therefore we have

IIR(z, A)xll-t 0

as

lzl-t oo,

Rez 2: a.

Since D(A) is dense in X and the resolvent of A is uniformly bounded on {z : D Rez 2: a}, this is true for all x EX and property (1.10) is proved. The following representation holds for every C0 -semigroup and is based on arguments from Fourier analysis, see van Neerven [204, Thm.l.3.3] and Kaashoek, Verduyn Lunel [139].

Theorem 3.8. LetT(·) be a Co-semigroup on a Banach space X with generator A. Then

1 T(t)x=2 (C,1) 7r =

1-oo 1-oo 00

1 27rt(C,1)

e 0 and x EX. Here, (C, 1) f~oo f(s) ds denotes the limit of the Cesaro means of f~r f(s)ds and is defined by

(C, 1)

I:

f(s) ds :=

J~oo ~faN

I:

f(s) dsdT,

whenever the limit on the right-hand side exists. Recall that Cesaro convergence of an integral is weaker than the existence of its principal value, and the latter is weaker than convergence. Proof. The second equality follows from integration by parts. To prove the first one take x EX and a> w 0 (T). By the representation

R(a+is,A)x =

1

00

e-iste-atT(t)xdt

and the Cesaro convergence of the inverse Fourier transform (see, e.g., Zaanen [261, Theorem 9.1] or Katznelson [146, Theorem VI.l.ll]) we obtain

1 e-atT(t)x= lim-N N --+00 271'

1N1T e.tR(a+is,A)xdsdT, 18

0

-T

and the first formula forT(·) follows for all a> wo(T). Take now an arbitrary a> s0 (A) and a 1 > max{a,w0 (T)}. By the previous considerations and Cauchy's theorem it suffices to show that

1

al

lim N--+oo

a

eb±iN

R(b ±iN, A)x db

=0

for every x EX. However, this follows directly from Lemma 3.7.

D

28

Chapter I. Functional analytic tools

We now consider a linear densely defined operator A satisfying so (A) < oo for which it is not known whether it is a generator or not. The basis of our approach is the following condition:

(R(a + i·, A) 2 x, y) E L 1 (JR.)

for all x EX, y EX',

(I.ll)

where a> s0 (A). Indeed, this property allows us to construct the inverse Laplace transform of the resolvent of the operator A which actually yields a semigroup which is strongly continuous on (0, oo ). (Note that strong continuity on (0, oo) does not imply strong continuity in general, see Kaiser, Weis [140].) The result is based on Shi, Feng [231] and Kaiser, Weis [140], see Eisner [64]. Theorem 3.9 (Laplace inversion formula). Let A be a densely defined linear oper•

ator on a Banach space X satisfying so(A) < oo and assume that condition (I.11) holds for all a > so (A). Then the bounded linear operators defined by T (0) = I and T(t)x := - 1 27r

1 1 00

. e(a+zs)t R(a +is, A)x ds

(I.12)

-00

00

=1 -

27ft _ 00

2 xds . e(a+zs)tR(a+is,A)

for all t > 0,

(I.13)

where the improper integrals converge in norm, are independent of a > so(A). In addition, the family (T(t) )t?.O is a semigroup which is strongly continuous on (0, oo) and satisfies lim T(t)x = x t---+0+

for all x E D(A 2 ).

(I.14)

Finally, we have R(z, A)x =

1

Proof. Define T(O)

00

for all x E D(A), Re z > so(A).

e-ztT(t)x ds

:=

I and

T(t)x := - 1 27r

1

00

. e(a+zs)t R(a +is, A)x ds

(I.15)

(I.16)

-00

for all x E X, t > 0 and some a > 0. We prove first that the integral on the right• hand side of (I.16) converges for all a > 0 and all x E X and does not depend on a> 0. For a fixed t > 0 and using fzR(z, A)= -R(z, A) 2 , we obtain for any

r > 0,

it J:r e(a+is)t R(a +is, A)x ds = e(a+ir)t R(a + ir, A)x- e(a-ir)t R(a- ir, A)x

+ i J:r e(a+is)tR(a+is,A) 2 xds.

3.

Specific concepts

i:

29

theory

from semigroup

i:

By (I.lO), the first two summands converge to zero if r

t

e(a+is)tR(a+is,A)xds =

--t

+oo. Therefore

e(a+is)tR(a+is,A) 2 xds,

(I.17)

and by condition (L11) the integral on the right-hand side converges. Indeed, for all r, R E IR, all x E X, and for B* = {y E X' : IIYII = 1} we have, by the uniform boundedness principle, that

1R

eist R(a +is, A) 2 x ds = sup

yEB*

1R (eist R(a +is, A) x, y) ds 2

r

holds for some constant L1(a) not depending on x. This implies the convergence of the integral on the right-hand side of (L17). Therefore the integral on the right-hand side of (I.16) converges and

1 00

. R(a +is, A) 2 xds e(a+ts)t T(t)x = 127rt _00

(I.18)

for every x E X and t > 0. We show next that T(t) does not depend on a Indeed, by Cauchy's theorem we obtain

J:r e(a+is)t R(a +is, A) 2 x ds=

-1b

> 0.

j_~ e(b+is)t R(b +is, A) 2 x ds

er+irR(r+ir,A) 2 xdr+

1b

er-irR(r-ir,A) 2 xdr

for all a, b > s0 (A). By (I.lO) and since a, b > so(A), the right-hand side converges to zero as r --t +oo. So we have proved that T(t) does not depend on a > 0 and formula (I.18) holds. From (1.18) we obtain

eat I(T(t)x,y)l::; 27rtii(R(a+i·,A) 2 x,y)lh

(1.19)

and, by the uniform boundedness principle, each T(t) is a bounded linear operator satisfying

ceat IIT(t)ll::; -t-,

t > 0,

(1.20)

for some constant C depending on a > s0 (A). We now show that (I.15) holds for all x E D(A). Take x E D(A), Re z > so(A) and a E (s 0 (A), Re z). Then, by Fubini's theorem and Cauchy's integral theorem

30

Chapter I. Functional analytic tools

for bounded analytic functions on a right half-plane, we have

00 1 100 e-zt 100 e(a+•s)tR(a+is,A)xdsdt . 1 e-ztT(t)xdt=0 2n 0

-oo roo e(a+is-z)tdt} R(a +is, A_)Ax +X ds

= .2_ 100 {

1

2n _ 00 } 0 a+2s = .2_ 00 R(a +is, A)Ax + x ds = R(z, A)Ax + x = R(z A)x 2n _ 00 (a+is)(z-a-is) z ' ' and (I.15) is proved. We next show strong continuity of our semigroup on (0, oo ). Since by (1.20) the semigroup is uniformly bounded on all compact intervals in (0, oo), it is enough to show that (I.14) holds for all x E D(A 2 ). (We used here that D(A 2 ) is dense, see e.g. Engel, Nagel [78, pp. 53-54]).) Take x E D(A 2 ), a> 0, and observe that

1

00 e(a+is)t (R(a +is, A)x- _x_. ) ds 2n _ 00 a+ 2s

T(t)x- X= .2_

= .2_ 1oo e(a+is)t R(a +is, A)Ax ds. a+ is

2n _ 00

Moreover, IIR(a +is, A)Axll :S gue's theorem,

11~~~~

for some constant c. Therefore, by Lebes•

1

. (Ttx-x () ) = 1- 00 R(a+is,A)Axds 11m t--->O+ 2n _ 00 a+ is

(1.21)

and the integral on the right-hand side converges absolutely. We now show

1

00 R(a + is,_A)Ax ds = O. _ 00 a+ 2s

(1.22)

Again by Cauchy's theorem and (I.lO) we have

111

r

-r

R(a+is,_A)Ax a+ 2s

dsll = 17r/2

irei'P R(a+rei'P,A)Axdcp -1r; 2 a+ re''P

:S11r 12 IIR(a+rei'P,A)Axlldcp---+0,

r---+oo.

-'lr/2

So equality (1.22) is proved, and (1.21) implies (I.14) and the strong continuity of our semigroup on (0, oo). We finally prove the semigroup law T(t + s) = T(t)T(s). Take t, s > 0, x E D(A 2 ), and 0 < .\ < f.l· Then R(A, A)x E D(A) and we have by (I.15)

R(f.l,A)R(.\,A)x=

1

00

e- 11 t{loo e->.sT(t)T(s)xds}dt.

3. Specific concepts from semigroup theory

31

On the other hand observe that

The uniqueness of the Laplace transform implies T(t+ s)x = T(t)T(s )x for almost all t, s > 0. By strong continuity ofT(-) on (0, oo) we have T(t + s)x = T(t)T(s)x for all t, s > 0 and x E D(A 2 ). Since D(A 2 ) is dense and the case t = 0 or s = 0 is trivial, the semigroup law is proved. D The following proposition shows that condition (I.ll) holds for a quite large class of semigroups and hence is indeed useful. The assertion in the Hilbert space case is based on van Casteren [45] and was used by many authors in this or some other form. Proposition 3.10. LetT(·) be a Co-semigroup on a Banach space X. If either X

is a Hilbert space or T(·) is an analytic semigroup, then the generator ofT(·) satisfies condition (I.ll). Proof. 1fT(·) is analytic, then the resolvent ofits generator A satisfies IIR(,\, A) II :::::; ~ for some M and all,\ in some sector I:c,cp := {z: arg(z- c):::::; r.p + r.p > 0.

af.!-:2

n,

Therefore, IIR 2 (a +is, A) II :::::; for every a > s(A) and s E JR., and condition (I.ll) follows. Assume now that X is a Hilbert space. Then by the representation

and Parseval's equality we have

for every x E X and every a > w0 (T). We now take a > so(A) and prove con• vergence of the above integral on the left-hand side. By the resolvent equality we have for a fixed a 1 > wo(T) and M := sup{JIR(z,A)II: Rez 2:: a} that IIR(a +is, A)xll

:S II(!+ (a1- a)R(a +is, A))R(a1 +is, A)xll :::::; (1 + (a1- a)M)IIR(al +is, A)xll

Chapter I. Functional analytic tools

32

I:

and hence

IIR(a +is, A)xll 2 ds

< oo

by the above considerations. Note that the same holds for the adjoint semigroup

T*(·) and its generator A* as well.

We now conclude that for every a Schwarz inequality

I:

I(R 2 (a +is, A)x, y)l ds:::;

: ; (I:

I:

1

IIR(a+is,A)xil 2 ds)

2

> s0 (A)

and x, y E X by the Cauchy•

IIR(a +is, A)xii·IIR(a- is, A*)yii ds

(I:

1

IIR(a+is,A*)yii 2 ds)

So the integrability condition (I.ll) is satisfied for every a > s0 (A).

2

< oo. 0

4 Positivity in £(H) As a preparation to Sections II.6 and III.7, we now look at the Banach algebra

.C(H), H a Hilbert space, decomposed as .C(H) = .C(H)sa EB i.C(H)sa, where .C(H)sa is the real vector space of all selfadjoint operators on H. We are interested in the order structure induced by the positive semidefinite operators on

.C(H)sa·

4.1

Preliminaries

Recall that an operator T E .C(H) on a Hilbert space H is positive semidefinite if (Tx, x) 2': 0 for every x E H, in which case we write 0 :::; T. The set .C(H)+ of all positive semidefinite operators is a generating cone in .C(H) and defines a vector space order by S :::; T if 0 :::; T - S. The following properties of this order will be needed later, see e.g. Pedersen [210, Proposition 3.2.9].

Lemma 4.1.

(a) IfO:::; S:::; T, then IISII:::; IITII·

(b) If {Sn}~=l is a positive, monotone increasing sequence in .C(H)sa such that sup IISnll < oo, nEN

then S := supnEN Sn exists and is given by (Sx, x) = supnEN(Snx, x) = limn--.oo (Snx, x). In particular, Sn converges to S in the weak operator topol• ogy.

4. Positivity in £(H)

33

Note that the last part of (b) follows using the polarisation identity. We now consider linear operators acting on £(H) respecting this order struc• ture.

Definition 4.2. An operator T E £(£(H)) is called positive (or positivity preserv• ing) if 0:::; TS for every 0:::; S E £(H). For a positive operatorS E £(£(H)) we again write 0:::; S. Analogously, we write

T:::; S whenever 0:::; S-T. Observe that such a positive operator is determined by its restriction to l(H)sa·

4.2

Spectral properties of positive operators

We now discuss some spectral and resolvent properties of positive operators on £(H). For the reader's convenience we present some proofs and start with the following lemma.

Lemma 4.3. Let H be a Hilbert space and letT be a positive operator on £(H). If IJLI > r(T), then 1\R(JL, T)Sx, x)l :::; (R(IJLI, T)Sx, x)

for every x

E

H and 0:::; S

E £(H).

Proof. For x E H, 0 :::; S E £(H) and JL satisfying IJLI Neumann series representation for the resolvent,

> r(T)

we have, by the

~ 1\TkSx,x)l ~ (TkSx,x) I lk+l = ~ I lk+l = (R(IJLI, T)Sx,x),

1\R(JL, T)Sx,x)l:::; ~ k=O

JL

k=O

JL

D

proving the assertion.

The following properties of positive operators on £(H) correspond to the classical Perron-Frobenius theorem for positive matrices and will be crucial for our approach to Lyapunov's equation in Sections II.6 and III. 7. Theorem 4.4. LetT be a positive operator on £(H), H a Hilbert space. Then the

following assertions hold.

(a) r(T)

E

CJ(T).

(b) R(JL, T) 2: 0 if and only if JL > r(T). In particular, r(T) < 1 if and only if 1 E p(T) and R(1, T) 2: 0. Proof. (a) For each,\ E CJ(T) such that 1,\1 = r(T) we have Jl--+>.,

lim

l11i>r(T)

IIR(JL, T)ll = oo.

By the uniform boundedness principle there exists S E £(H)+ and a sequence {JLn}~=l converging to ,\ such that limn--+oo IIR(JLn, T)SII = oo. This implies, by

Chapter I. Functional analytic tools

34

the polarisation identity, that limsupn_,oo I(R(JLn, T)Sx, x)l = oo for some x E H. We obtain, by Lemma 4.3, limsup(R(IJLnl, T)Sx, x) = oo n->oo

implying l.\1 = r(T) E (J"(T). (b) The "if" direction follows directly from the Neumann series representation for the resolvent of T. To show the "only if" direction, assume that R(JL, T) 2': 0 holds for some JL E p(T). We first show that JL is real. Take some v > r(T) with v # JL. By the above "if" direction, R(v, T) 2': 0. For every x E H we obtain by the resolvent identity

(JL - v) (R(JL, T)R(v, T)I x, x) = (R(v, T)I x, x) - (R(JL, T)I x, x)

ER

Since v # JL and hence R(v, T)I # R(JL, T)I, we have (R(v, T)Ix, x) # (R(JL, T)Ix, x) for some x E H. Thus the right-hand side of the above equation is real and non-zero for this x and hence R(JL, T)R(v, T) 2': 0 implies JL E R To show JL > r(T), we assume JL :::; r(T) and take v > r(T). By the above, R(v, T) 2': 0 and therefore, by the resolvent identity,

R(JL, T) = R(v, T) + (v- JL)R(JL, T)R(v, T) 2': R(v, T).

(1.23)

By (a) we have limv->r(T)+ IIR(v, T)ll = oo implying by the uniform boundedness principle that limsupv->r(T)+ IIR(v, T)SII = oo for someS E .C(H)+· Moreover, it follows from the polarisation identity that lim supv->r(T)+ (R(v, T)Sx, x) = oo for some x E H, contradicting (1.23). D

4.3 Implemented operators We now present an important class of positive operators on .C(H) which will play a crucial role in Section 11.6.

Definition 4.5. To T E .C(H), H a Hilbert space, corresponds its implemented operator Ton .C(H) defined by TS := T* ST, S E .C(H). Implemented operators have the following elementary properties.

Lemma 4.6. For each T E .C(H), the implemented operator T is positive and satisfies II Til = II Till = IITII 2 .

Proof. Positivity of T follows directly from its definition. For the second part observe that IITSII = liT* STII :S IITII 2 IISII for every S implies II Til :S IITII 2 . On the other hand we have IITII

2':

liTIll= IIT*TII = IITII 2 , proving the assertion.

D

4. Positivity in C(H)

35

4.4 Implemented semigroups Starting from a Co-semigroup on a Hilbert space, we define its "implemented semigroup" and list some basic properties.

Definition 4.7. For a Co-semigroup T(·) on a Hilbert space H, we call the family (T(t))t~o of operators on C(H) given by

T(t)S := T*(t)ST(t),

S E .C(H),

the implemented semigroup corresponding toT(·). The family T(·) is indeed a semigroup, but the mapping t f--+ T(t)Sx is only continuous for every S E .C(H) and x E H, i.e., T(·) is "strongly operator continuous". It is a C0 -semigroup on C(H) if and only if T(·) is norm continuous. The following properties ofT(·) are well-known, see e.g. Nagel (ed.) [196, Section D-IV.2], Batty, Robinson [28, Example 2.3.7] and also Kiihnemund [158].

Lemma 4.8. For the semigroup T(-) on C(H) implemented by a C0 -semigroup T(·) on a Hilbert space H the following assertions hold. (a) IIT(t)ll = IIT(t)III = IIT(t)ll 2 · (b) There is a unique operator A with domain D(A) in C(H), called the gener• ator ofT(·), with the following properties. 1) D(A) is the set of all S E .C(H) such that S(D(A)) c D(A*) and the operator A* S +SA : D(A) ----t H has a continuous extension to H

denoted by AS.

J

2) If 000 e->.tT(t)Sxdt converges for every S E .C(H) and x E H, then A E p(A) and

R(A, A)Sx

=

loo e->.tT(t)Sx dt.

In particular, this equality holds for every A with ReA > wo (T).

To shorten the notation, we write R(A, A) = J000 e->.tT(t) dt meaning that the integral exists in the above sense. While T(·) is not strongly continuous on C(H) in general, we gain posi• tivity as a new property. Indeed, T(·) is positive on C(H) since (T(t)Sx,x) = (ST(t)x, T(t)x) ~ 0 for every positive semidefinite operator S E .C(H) and every

xEH. Such semigroups have been studied systematically in e.g. Nagel (ed.) [196, Section D-IV.1-2], Nagel, Rhandi [199], Groh, Neubrander [112] and Batty, Robin• son [28, Section 2]. For example, the following spectral property is analogous to Theorem 4.4 above and plays the key role in our considerations.

Theorem 4.9. Let (T(t))t~o be an implemented semigroup on C(H) for a Hilbert space H with generator A. Then s(A) E a( A), and R(A, A) ~ 0 in C(H) if and only if A > s(A). Moreover, wo(T) = s(A) holds.

Chapter II

Stability of linear operators In this chapter we study power and polynomial boundedness, strong, weak and almost weak stability of linear bounded operators on Banach spaces. In many cases, these properties can be characterised through the behaviour of the resolvent of the operator in a neighbourhood of the unit circle. Similar results on the stability of C0-semigroups will be discussed in Chapter III.

1 Power boundedness We start by discussing power boundedness and the related property of polynomial boundedness of an operator on a Banach space. While power boundedness is fun• damental for many purposes, it is difficult to check in the absence of contractivity. On the contrary, polynomial boundedness is much easier to characterise.

1.1 Preliminaries We first introduce power bounded operators and show some elementary properties.

Definition 1.1. A linear operator Ton a Banach space X is called power bounded if SUPnEN IITnll < 00. An immediate necessary spectral condition for power bounded operators is the following.

Remark 1.2. The spectral radius r(T) of a power bounded operator T on a Banach space X satisfies

r(T) = inf (IITnll)~ :S 1,

nEN

and hence a(T) C { z :

lzl :S 1}.

38

Chapter II. Stability of linear operators Note that r(T) :::;: 1 does not imply power boundedness as can be seen from

T =

GU

on C 2 . Moreover, we refer to Subsection 1.3 for more sophisticated

examples and a quite complete description of the possible growth of the powers of an operator satisfying r(T) :::;: 1, see Example 1.16 below. However, the strict inequality r(T) < 1 automatically implies a much stronger assertion. Proposition 1.3. Let X be a Banach space and T E L:(X). The following assertions are equivalent.

(a) r(T) < 1. (b) limn->oo IITnll = 0. (c) T is uniformly exponentially stable, i.e., there exist constants M > 0 and c > 0 such that IITnll:::;: Me-en for all n EN. The proof follows from the formula r(T) = limnEN(IITnll)*.

Remark 1.4. It is interesting that for power bounded operators on separable Ba• nach spaces some more information on the spectrum is known. For example, Jami• son [135] proved that for every power bounded operator T on a separable Banach space X the boundary point spectrum Pa(T) n r is countable. For more infor• mation on this phenomenon see Ransford [218], Ransford, Roginskaya [219] and Badea, Grivaux [13, 14]. We will see later that countability of the entire spectrum on the unit circle plays an important role for strong stability of the operator (see Subsection 2.3). The following easy lemma is useful in order to understand power boundedness. Lemma 1.5. Let T be power bounded on a Banach space X. Then there exists an equivalent norm on X such that T becomes a contraction. Proof. Take llxll1 := supnENU{O} IITnxll for every x EX.

D

Remark 1.6. It is difficult to characterise those power bounded operators on Hilbert spaces which are similar to a contraction for a Hilbert space norm. Foguel [87] showed that this is not always true (see also Halmos [120]). The next result, due to Guo, Zwart [113] and van Casteren [46], characterises power bounded operators by strong Cesaro-boundedness of the orbits. Proposition 1.7. An operator Ton a Banach space X is power bounded if and only if

1 n sup-11Tkxll 2 nEN n + 1 k=O

L

< oo

for every x EX,

1 n sup--""' IIT'kyll 2 nEN n+ 1 ~

< oo

for every y EX'.

k=O

39

1. Power boundedness

Proof. The necessity of the above conditions is clear, so we have to prove suffi• ciency. Fix x EX andy EX'. By the Cauchy-Schwarz inequality we obtain I(Tnx, y)l = n: 1

.C

1

Tk

k=O

=

1 ~ (k (n- 1)! L...t

+ 1) · · · (k + n- 1) Tk >,k+n

k=O

This implies IIRn(>.. T)ll

'

< M - (n- 1)!

f

.

(k + 1) ... (k + n -1) l>..lk+n

k=O

t;

=

M(- 1)n-1 (n _ 1)!

=

M(-1)n-1 ( (n- 1)!

oo

(

1 )(n-1) zk+1

lz=l>.l

1 )(n-1) Z- 1 lz=l>.l

=

M (1>..1- 1)n'

0

and the proposition is proved.

It is surprising that the converse implication in Proposition 1.10 is not true, i.e., the discrete analogue of the Hille-Yosida theorem does not hold. For a coun• terexample with the maximal possible growth of IITnll being equal to ..,fii, see Lubich, Nevanlinna [172]. For a systematic discussion of the strong Kreiss con• dition and the related uniform Kreiss condition, we refer the reader to Gomilko, Zemanek [106], Montes-Rodriguez, Sanchez-Alvarez and Zemanek [187], see also Nagy, Zemanek [200] and Nevanlinna [206]. Note that the question whether the strong Kreiss resolvent condition implies power boundedness for operators on Hilbert spaces is still open. Condition (II.l) for n = 1 is called the Kreiss resolvent condition and plays an important role in numerical analysis. By the celebrated Kreiss matrix theorem, it is equivalent to power boundedness for operators acting on finite-dimensional spaces. On infinite-dimensional spaces this is no longer true, see Subsection 1.3 for details. We also mention here the Ritt resolvent condition (also called Tadmor-Ritt

resolvent condition) IIR(>.., T)ll

~ l~o:s~l

for all >.. with 1>..1

> 1.

This condition does imply power boundedness, see Nagy, Zemanek [200], and also n r c {1 }, hence it is far from being necessary for power boundedness of general operators. We refer to Shields [232], Lubich, Nevanlinna [172], Nagy, Zemanek [200], Borovykh, Drissi, Spijker [40], Nevanlinna [206], Spijker, Tracogna,

a(T)

1. Power boundedness

41

Welfert [233], Tsedenbayar, Zemanek [241] and Vitse [246H248] for systematic studies of operators satisfying Kreiss and Ritt resolvent conditions. In the following we consider conditions not involving all powers of the resol• vent and characterise power boundedness at least on Hilbert spaces. We first state an easy but very useful lemma. Lemma 1.11. Let X be a Banach space and T E .C(X). Then

~11~ eicp(n+l)R(rei'P,T)dcp = Tn = _r_ 21r

for every n

0

E

r ~2

21r(n + 1)

1~ eicp(n+l)R 2(rei'P,T)dcp 0

(II.2)

N and r > r(T).

Proof. The first equality in (II.2) follows directly from the Dunford functional calculus and the second by integration by parts. D The main result of this subsection is the following theorem which is a discrete analogue of a characterisation of bounded Co-semigroups due to Gomilko [104] and Shi, Feng [231], see Theorem III.l.ll. Theorem 1.12. Let X be a Banach space and T E .C(X) with r(T) ::; 1. Consider

the following assertions. (a) limsupr__,l+(r- 1) J;n IIR(rei'P, T)xll 2dcp

< oo for all x EX, 2 limsupr__,l+(r -1) J;n IIR(rei'P,T')yll dcp < oo for ally EX';

(b) lim supr--+1+ (r - 1) J02n I(R 2 (rei'P, T)x, y) ldcp

< oo for all x

E

X, y E X';

(c) T is power bounded. Then (a)=?(b)=?(c). Moreover, if X is a Hilbert space, then (c)=?(a), hence (a), (b) and (c) are equivalent. Proof By the Cauchy-Schwarz inequality we have

fo i(R 2(rei'P, T)x, y) ldcp = 2

:S:

1I

2 n (R(reicp, T)x, R(reicp, T')y) ldcp

(1

2 niiR(rei'P, T)xll 2dcp)

1

2

(1

2 niiR(rei'P, T')vll 2

dcp)

1

2

for all x EX, y EX' and r > r(T). This proves the implication (a)=?(b). For the implication (b)=?(c) taken EN and r > 1. By Lemma 1.11 we have

rn+2 {2n I(Tnx, Y)l :S: 21r(n + 1) Jo I(R 2(rei'P)x, y)ldcp for every x EX andy EX'. By (b) and the uniform boundedness principle there exists a constant M > 0 such that

{21r

(r -1) Jo

I(R 2(rei'P,T)x,y)ldcp::; MllxiiiiYII for every xEX, yEX' and r>l.

Chapter II. Stability of linear operators

42 Therefore we obtain

(11.3) Take now r := 1 + n+l. Then (n+l)(r- 1) = 1 + n+l obtain by (11.3) that supnEN IITnll is finite. rn+2

1

(

1

)n+2

---+ e

as n---+ oo, and we

For the second part of the theorem assume that X is a Hilbert space and T is power bounded. Then, by Lemma 1.11 and Parseval's equality,

(r -1) {27r }o

IIR(rei'~', T)xll2d

1,

which implies 2 2 (r-1) { triiR(rei'P,T)lll 2d'P = (r-1) { tr(

Jo

1 ) d'P =

Jo r- 1 2

~ r- 1

--7

oo

as r --71+.

This shows that condition (a) in Theorem 1.12 does not hold for T. In view of Remark 1.13.2) we note also that limr___,I+(r- 1) J~tr IIR(rei'P, T)xiiPd'P = oo for every 1 < p < oo. The pre-adjoint ofT on l 1 also does not satisfy (a). We remark that this operator is again isometric and invertible. A useful characterisation of power boundedness on Banach spaces is still unknown.

1.3 Polynomial boundedness In this subsection we discuss the related notion of polynomial boundedness which, surprisingly, is much easier to characterise. Definition 1.15. A bounded linear operator T on a Banach space X is called polynomially bounded if IITnll ::; p(n) for some polynomial p and all n EN. Without loss of generality we will assume the polynomial to be of the form

p(t) = Ctd.

Note that a polynomially bounded operator T again satisfies r(T) ::; 1. The following example shows that the converse implication is far from being true. Example 1.16 (Operators satisfying r(T) ::; 1 with non-polynomial growth). Con• sider the Hilbert space H

:~ 1; ~ { {xn}~ 1 C C:

t, lxnl2a~


0 and all n EN.

Moreover, if (II.9) holds ford= k, then (II.8) holds with d = k + 1.

(II.9)

2. Strong stability

45

Proof. Assume that condition (II.8) holds and taken EN and r > 1. By Lemma 1.11 and (II.8) we have

forM> limsuplzi-+I+(Izl-1)diiR(z, T)ll and r close enough to 1. Taking r := 1+~ for n large enough we obtain IITnll ::; 2M end and the first part of the theorem is proved. For the second part we assume that condition (II.9) holds for d = k. Take n EN, r > 1, r.p E [0,2rr), and q := ~ < 1. Then oo k-1 oo oo IITnll nkqn nk + C nkqn::; C rn+ 1 :S Cq n=k n=O n=O n=O C6k' k-1 dk 00 k-1 ::; C ~ nk + CCqk dqk ~ qn ::; C ~ nk + (1 - q)~+l'

IIR(rei"', T)ll :S

L

L

L

L

where 6 is such that nk ::; 6 · n(n- 1) · ... · (n- k + 1) for all n 2: k. For k = 0 we suppose the first sum on the right-hand side to be equal to zero. Substituting D q by ~ we obtain condition (II.8) for d = k + 1. Remark 1.18. Note that, by the inequality dist(A, a(T)) 2: IIR(I,T)II, condition (II.8) for 0 ::; d < 1 already implies r(T) < 1 and hence uniform exponential stability. So for 0 ::; d < 1 Theorem 1.17 does not give the best information about the growth of the powers. Nevertheless, for d = 1, i.e., for the above mentioned Kreiss resolvent condition, the growth stated in Theorem 1.17 is the best possible and the exponent din (11.9) cannot be decreased in general, see Shields [232]. For d > 1 it is not clear whether Theorem 1.17 is optimal.

2 Strong stability In this section we consider a weaker stability concept than the norm stability discussed in Proposition 1.3 and replace uniform by pointwise convergence.

2.1 Preliminaries Let us introduce strongly stable operators and present some of their fundamental properties.

Definition 2.1. An operator T on a Banach space X is called strongly stable if limn-+oo IITnxll = 0 for every x E X. The first part of the following example is, in a certain sense, typical for Hilbert spaces (see Theorem 2.11 below).

46

Chapter II. Stability of linear operators

Example 2.2.

(a) Consider H := l2 (N, Ho) for a Hilbert space H0 and T E £(H)

defined by

T(x1, x2, x3, .. .) := (x2, x3, .. .).

(II.lO)

The operator T, called the left shift on H, is strongly stable. Analogously, the operator defined by formula (II.lO) is also strongly stable on the spaces co(N, X), ZP(N, X) for a Banach space X and 1 ~ p < oo, but not on l 00 (N, X). (b) Consider X := [P for 1 ~ p < oo and the multiplication operator T E £(X) defined by T(xl,X2,X3,···) := (alx1,a2X2,a3x3,···) for a sequence (an)~=l C {z: lzl < 1}. By the density of vectors with finitely many non-zero coordinates, the operator T is strongly stable. Note that by Proposition 1.3, T fails to be uniformly exponentially stable if supn lanl = 1. Analogously, such a multiplication operator is also strongly stable on c0 but not on zoo. The following property of strongly stable operators is an easy consequence of the uniform boundedness principle. Remark 2.3. Every strongly stable operator T on a Banach space X is power bounded which in particular implies u(T) C { z: lzl ~ 1}. Moreover, the properties Pa(T) n r = 0 and Pa(T') n r = 0 are necessary for strong stability.

We now present an elementary property which is very helpful to show strong stability. Lemma 2.4. Let X be a Banach space, T

E £(X) power bounded and x EX.

(a) If there exists a subsequence {nk}~ 1 C N such that limk_,oo IITn•xil = 0, then limn->oo IITnxll = 0. (b) IfT is a contraction, then there exists limn->oo IITnxll·

Proof. The second assertion follows from the fact that for a contraction the se• quence {IITnxll}~=l is non-increasing. For the first one take E > 0, M := supnEN IITnll and kEN such that IITnkxll ~E. Then we have

for every n 2:: nk, and (a) is proved.

D

Remark 2.5. Assertion (b) in the above lemma is no longer true for power bounded operators. Here is an example. Consider the Hilbert space of alll 2-sequences en• dowed with the norm

47

2. Strong stability

On this space consider the right shift operator which is clearly power bounded. We see that for the vector e1 = (1, 0, 0, ... ) we have IIT 2n-le1ll = ~ and IIT 2ne 1ll = 1 for every n E N which implies

The following is a direct consequence of Lemma 2.4. Corollary 2.6. A power bounded operator T on a Banach space X is strongly stable

if and only if

for every x in a dense subset of X. One can also characterise strong stability without using power boundedness. This is the following result due to Zwart [263]. Proposition 2.7. An operator Ton a Banach space X is strongly stable if and only

if

1

n

lim - - " 11Tkxll 2 = 0 for all x E X and

n + 1 L..J

n->oo

k=O

1 sup - -1 nEN

(II.ll)

n

+

L IIT'kyll n

2

< oo for ally E X'.

(II.12)

k=O

The proof is analogous to the one of Proposition 1. 7. Note that one can formulate Corollary 2.6 and Proposition 2.7 for single orbits. More precisely, limn->oo rnx = 0 if and only if conditions (11.11) and (11.12) hold for every y E X'. In particular, for a power bounded operator and x E X, limn-->00 rnx = 0 is equivalent to (11.11). Finally, we state a surprising and deep result of Miiller [188] on the asymp• totic behaviour of operators which are not uniformly exponentially stable. Theorem 2.8 (V. Miiller). LetT be a bounded linear operator on a Banach space X with r(T) 2: 1. For every {an} ~=l C [0, 1) converging monotonically to 0 there exists x E X with llxll = 1 such that IITnxll 2: an

for every n E N.

In other words, the orbits of a strongly but not uniformly exponentially stable operator decrease arbitrarily slowly. We refer to Miiller [188], [189] and [191, Section V.37] for more details and results. Up to now, there is no general characterisation of strong stability for opera• tors on Banach spaces, while on Hilbert spaces there is a resolvent condition (see Subsection 2.4).

Chapter II. Stability of linear operators

48

2.2

Representation as shift operators

We now present a characterisation analogous to Theorem 1.8 of strongly stable operators as shifts. Proposition 2.9. Let T be a strongly stable contraction on a Banach space X. Then T is isometrically isomorphic to the left shift on a closed subspace of c0 (X).

This result follows directly from Theorem 1.8. Moreover, by the renorming procedure we again obtain a characterisation for non-contractive strongly stable operators. Corollary 2.10. A strongly stable operator T on a Banach space X is isomorphic to the left shift on a closed shift-invariant subspace of co(Xl), where X 1 is X endowed with an equivalent norm.

For contractions on Hilbert spaces we have the following classical result of Foia§ [85] and de Branges-Rovnyak [43] (see also Sz.-Nagy, Foia§ [238, p. 95]). Theorem 2.11. LetT be a strongly stable contraction on a Hilbert space H. Then T is unitarily isomorphic to a left shift, i.e., there is a Hilbert space H 0 and a unitary operator U : H _, H 1 for some closed subspace H 1 c l 2 (N, H 0 ) such that UTU- 1 is the left shift on H1.

Proof. As in Proposition 2.9, the idea is to identify a vector x with the sequence

{Tnx }~=D in an appropriate sequence space. Observe first that by strong stability we have the equality 00

llxll = 2)11Tnxll 2 -11Tn+ 1 xll 2 ). 2

(II.l3)

n=O

We now define on H a new seminorm by

corresponding to the scalar semiproduct (x, y)y := (x, y) - (Tx, Ty). Note that contractivity ofT implies the non-negativity of II·IIY· Let Ho := {x: llxiiY = 0} and finally take the completion Y := (H / Ho, I · IIY f. Define now the operator J : H _, l 2 (Y) by

which identifies x with its orbit under (Tn)::'=o· By (II.13), J is an isometry and hence a unitary operator from H to its (closed) range. Observe finally that the operator JT J- 1 acts as the left shift on rg J. 0

49

2. Strong stability

2.3 Spectral conditions In this subsection we discuss sufficient conditions for strong stability of an operator in terms of its spectrum. All results presented here are based on some "smallness" of the part of the spectrum on r. The origin of this spectral approach to stability is the following classical "T = I" theorem of Gelfand [95]. The proof we give here is due to Allan, Ransford

[6].

Theorem 2.12 (Gelfand, 1941). Let X be a Banach space and T E .C(X). If a-(T) = {1} and supnEZ IITnll < oo, then T = I.

Proof. Since (the principle branch of) the logarithm is holomorphic in a neigh• bourhood of CJ(T) = { 1}, we can define S := -i log T using Dunford's functional calculus. This operator satisfies T = eiS and CJ(S) = 0 by the spectral mapping theorem. We show that S = 0. Taken E N and consider sin(nS) := 1i(einS- e-inS) = 1i(Tn- y-n). We observe that this operator is power bounded satisfying ll[sin(nSWII =

(

Tn _ y-n)k .

2z

~sup IITnll nEZ

for every kEN.

Consider now the Taylor series representation I:;;o=O CkZk of the principal branch of arcsin in 0 (recall that CJ(nS) = {0} ). Since Ck 2 0 and L~o Ck = arcsin(1) = ~' we obtain by the above that

llnSII = II arcsin(sin(mS))II

~

L Ck II (sin(nS))k II ~ ~sup IITnll nEZ 00

k=O

This implies S = 0 and T = eiS = I.

for every nEN. D

As a second tool, we introduce the so-called isometric limit operator, an elegant construction due to Lyubich, Vu [180]. For a contraction T on a Banach space X define

Y := {x EX: lim IITnxll = 0}. n-+oo

Observe that Y is a closed T-invariant subspace. Consider the space (X/Y, ll·llr) for the norm llx + Yllr := limn-+oo IITnxll which is well defined by Lemma 2.4. Note that llx+ Yllr ~ llxll holds for every x EX by contractivity ofT. Define the operatorS on X/Y by S(x + Y) := Tx + Y. Consider finally the completion Z := (X/Y, II ·llr

r.

50

Chapter II. Stability of linear operators

Remark 2.13. The space (X/Y, ll·llr) is not complete in general. This can be seen taking the space X= {(xk)~-oo E l1 (Z): 2.:~= 1 klxkl < oo} with norm given

by ll(xk)':'ooll := 2.:~=-oo lxkl is contractive and satisfies

+ 2.:~= 1 klxkl

and the left shift operator T. Then T

llxllr = lim IITnxll = llxlh

n->oo

Therefore we have (X/Y, II . llr)

= (X, II . Ih)

for every x E X. and its completion is

z = l 1 (Z).

We now need the following lemma, see Lyubich, Vii [180] or Engel, Nagel [78, pp. 263-264] in the continuous case. Lemma 2.14. LetT be a contraction on a Banach space X and takeS on XjY as above. Then S extends to an isometry on Z, called the isometric limit operator, and satisfies O"(S) C O"(T).

Proof. Observe first that IIS(x + Y)llr = limn_, 00 11Tn+ 1xll = llxllr and hence S extends to an isometry on Z. Take now A E p(T) and define operators R(A) on (XjY, II· llr) by R(A)(x + Y) := R(.\, T)x + Y,

x EX.

Since llx + Yllr :"::: llxll for every x EX, R(A) extends to a bounded operator on Z again denoted by R(A). By definition we have (A-S)R(A) =I and R(A)(A-S) =I on XjY and hence A E p(S) and R(A) =(A- S)- 1 . 0 The following theorem is the starting point for many spectral characterisa• tions of strong stability, see Katznelson, Tzafriri [147]. Our proof of the non-trivial implication is due to Vii [249]. Theorem 2.15 (Katznelson-Tzafriri, 1986). LetT be a power bounded operator on a Banach space X. Then IITn+l_Tnll----; 0 as n----; oo if and only ifO"(T)nr C {1}.

Proof. Assume first that there exists 1 -1- A E r with A E O"(T). Then we have, by the spectral mapping theorem for polynomials,

and hence liminfn->oo 11Tn+ 1 - Tnll

> 0.

To show the converse implication, we first show that the condition O"(T) n r C {1} and the power boundedness ofT imply lim 11Tn+ 1x- Tnxll = 0

n->oo

for every x E X.

(11.14)

By Lemma 1.5 we may assume that Tis a contraction. Let S be its isometric limit operator. Then S is an isometry with O"(S) n r c {1} by Lemma 2.14. Since an isometry is non-invertible if and only if its spectrum is the entire unit disc, see, e.g., 1 being an isometry Conway [52, Exercise VII.6.7], S must be invertible with

s-

51

2. Strong stability

as well. This shows that a(S) = {1}, and by Gelfand's Theorem 2.12, S =I on Z. By the definition of S, we have that x- Tx E Y = {y EX: limn--+oo IITnyll = 0}, and (11.14) is proved. It remains to show that limn--+oo IITn+l- Tnll = 0. Consider the operator U on .C(X) given by UR = T R for all R E .C(X). This operator is power bounded and satisfies a(U) C a(T). By the above considerations we have limn--+oo IIUn+l R• Rll = 0 for all R E .C(X). It suffices to takeR:= I. D

un

For alternative proofs of the Katznelson~ Tzafriri theorem using complex analysis see Allan, Ransford [6] and Allan, Farrell, Ransford [7], as well as the original proof of Katznelson, Tzafriri [147] using harmonic analysis.

Remark 2.16. There are various generalisations and extensions of the Katznelson~ Tzafriri theorem in which rn+I- rn is replaced by rn f(T) for some function f. For example, Esterle, Strouse, Zouakia [80] showed that for a contraction T on a Hilbert space and for f E 1t(l!}) n C(f), the condition limn--+oo IITnf(T)II = 0 holds if and only iff = 0 on a(T) n r. This can be generalised to "polynomially bounded" operators (in the sense of von Neumann's inequality, see, e.g., Pisier [213, 214] and Davidson [56]) and to holomorphic functions on II} and completely non• unitary contractions, see Kerchy, van Neerven [148] and Bercovici [30], respectively. The version of the Katznelson~Tzafriri theorem for C0 -semigroups is in Esterle, Strouse, Zouakia [81] and Vu [250]. For further generalisations and history we refer to Chill, Tomilov [50, Section 5]. An immediate corollary concerning strong stability is the following.

Corollary 2.17. Let T be a power bounded operator on a Banach space X with a(T)nr C {1 }. Then IITnxll --+ 0 as n--+ oo for every x E rg(I- T). In particular, 1 ~ Per (T') implies that T is strongly stable. On the basis of the Katznelson~ Tzafriri theorem, Arendt, Batty [9] and in• dependently Lyubich, Vu [180] proved a sufficient condition for strong stability assuming countability of a(T) on the unit circle. The following proof using the isometric limit operator is due to Lyubich and Vu [180], while Arendt, Batty [9] used Laplace transform techniques.

Theorem 2.18

(Arendt~Batty~Lyubich~Vu,

1988). LetT be power bounded on a

Banach space X. Assume that

(i) Pa(T') n r = 0; (ii) a(T)

nr

is countable.

Then T is strongly stable. Proof We first note that, by Lemma 1.5, we can assume T to be contractive. Assume that T is not strongly stable. Then the space z = (XjY, II . llr constructed above is non-zero. Consider now the isometric limit operator Son Z. By (ii) and Lemma 2.14, a(S) n r is still countable implying a(S) n r # r. Since

r

Chapter II. Stability of linear operators

52

the spectrum of a noninvertible isometry is the whole unit disc, S is invertible and a(S) C r holds. So a(S) is a countable complete metric space and therefore contains an isolated point Ao E f by Baire's theorem. We now show that >.0 is an eigenvalue of S. Consider the spectral projection P of S corresponding to >.o and the restriction S0 of S on rg P =: Z 0 . Since a(So) = {Ao} and IISoll = 1 for every n E Z, it follows from Gelfand's theorem (Theorem 2.12) that S0 = >. 0 1 and therefore in particular >.0 E Pa(S). It remains to show that >.0 E Pa(T') to obtain a contradiction to (i). Take z' E Zb and consider 0 i= x' E X' defined by

(x, x')

:=

(P(x

+ Y), z'),

x EX.

Then

(x, T' x') = (Tx, x') = (P(Tx + Y), z') = (PS(x + Y), z') = >.o(P(x + Y), z') = >.o(x, x') for every x EX and hence T'x' =>.ox' contradicting (i).

D

Remark 2.19. For operators with relatively weakly compact orbits (in particular, for power bounded operators on reflexive Banach spaces) condition (i) is equivalent to Pa(T) nr = 0.

As one of many applications of the above theorem we present the following stability result for positive operators. Corollary 2.20. Let T be a positive power bounded operator on a Banach lattice.

Then T is strongly stable if Pa(T') n r = 0 and a(T) n f

i=

f.

The proof follows from Theorem 2.18 and the Perron-Frobenius theory stat• ing that in this case the boundary spectrum a(T) nr is multiplicatively cyclic (see Schaefer [227, Section V.4]), and hence, since different from r, must be a finite union of roots of unity. The following result is an extension of the Arendt-Batty-Lyubich-Vii theo• rem for completely non-unitary contractions on Hilbert spaces (for the definition of completely non-unitary operators see Remark 3.10) below. Theorem 2.21 (Foi8§, Sz.-Nagy [238, Prop. 11.6.7]). LetT be a completely non•

unitary contraction on a Hilbert space H. If a(T) n r has Lebesgue measure 0, then T and T* are both strongly stable. See also Kerchy, van Neerven [148] for related results. In the above theorem it does not suffice to assume contractivity and emptiness of the point spectrum. In Example III.3.19 we present a unitary and hence not strongly stable operator T satisfying >.( a(T)) = 0 for the Lebesgue measure >. and

Pa(T) = 0.

53

2. Strong stability

Remark 2.22. Although the results above have many useful applications, the small• ness conditions on the boundary spectrum are far from being necessary for strong stability. Indeed, we saw in Example 2.2(b) that for a strongly stable operator the boundary spectrum can be an arbitrary closed subset of r. However, on super• reflexive Banach spaces, countability of the boundary spectrum is equivalent to a stronger property called superstability, see Nagel, Riibiger [198].

2.4 Characterisation via resolvent In this subsection we pursue a resolvent approach to stability introduced by Tomilov [243]. Our main result is the following discrete analogue to the spectral character• isation given by Tomilov. For related results we refer to his paper [243].

Theorem 2.23. Let X be a Banach space and T E £(X) with r(T) ::; 1 and x E X. Consider the following assertions. (a) lim (r-1) 1

211"

IIR(rei'~',T)xll 2 dtp=O and

01211"

r-+1+

limsup(r- 1) r-+1+

0

IIR(rei'~', T')yll 2 dtp

< oo for ally EX'.

(b) lim IITnxll = 0. n-+oo

Then (a) implies (b). Moreover, if X is a Hilbert space, then (a)#(b). In particular, condition (a) for all x E X implies strong stability ofT and, in the case of a Hilbert space, is equivalent to it. Proof. To prove the first part of the theorem we take x E X, n E N, and r > 1. By Lemma 1.11 and the Cauchy-Schwarz inequality we have I(Tnx, y) I :S:

, 1.

(The uniform boundedness principle should be applied to the family of operators Sr : X' --t L 2 ([0,27r],X'), r > 1, given by (Sry)(r.p) := ..;r="'R(rei'~',T')y.) Therefore, we obtain

(II.15)

Chapter II. Stability of linear operators

54 rn+2

.

1

For r := 1 + n+ 1 , we obtam (n+l)(r- 1) =

(

1+

1 ) n+2

n+l

----+ e as n ----+ oo, hence

limn--+oo IITnxll = 0 by (II.15). Assume now that X is a Hilbert space and T is strongly stable. Then by Lemma 1.11 and Parseval's equality,

for s := ~ < 1. The right-hand side is, up to the factor 1/(r + 1), the Abel mean of {11Tnxll 2 }. Therefore it converges to zero ass----+ 1 by the strong stability ofT, proving the first part of (a). The second part of (a) follows from Theorem 1.12. 0 We now show that the converse implication in Theorem 2.23 does not hold on Banach spaces. We use ideas as in Example 1.14.

Example 2.24. Consider the space 100 and the multiplication operator T given by

for a sequence {an}~=l C {z: lzl and x E U1(l) we obtain

< 1} satisfying

r

C {an: n EN}. For I.AI

>1

2

IIR(.A, T)xll

lxnl

=~~~I .A- ani :::::

1 2dist(.A, r)

1

2(1>.1-1)

and therefore

rrr IIR(ret'~',T)xll . dcp 2: 2n(r-1) (r _ 1)2 = 2(r _ 1)----+ oo 1f

2

(r -1) Jo

asr----+1+.

Consider now X = l 1 and the multiplication operator S defined by the se• quence {an}~=l· Then S is strongly stable, see Example 2.2 (b). On the other hand, since S' = T and by the considerations above, the second part of (a) in Theorem 1.12 is not fulfilled for an open set in X'. By Theorem 1.12 we immediately obtain the following characterisation of strongly stable operators on Hilbert spaces. Corollary 2.25. Let T be a power bounded operator on a Hilbert space H and x EX. Then IITnxll----+ 0 if and only if

lim (r -1) {

r--+1+

Jo

2

'/r

IIR(rei'~',T)xll 2 dcp = 0.

(II.16)

In particular, T is strongly stable if and only if (II.16) holds for every x E H (or only for every x in a dense subset of H). It remains an open question whether the assertion of Corollary 2.25 holds on arbitrary Banach spaces. More generally, it is not clear what kind of resolvent conditions characterise strong stability of operators on Banach spaces.

3. Weak stability

55

3 Weak stability We now consider stability of operators with respect to the weak operator topology. Surprisingly, this is much more difficult to characterise than the previous concepts.

3.1

Preliminaries

We begin with the definition and some immediate properties of weakly stable operators.

Definition 3.1. Let X be a Banach space. An operator T E .C(X) is called weakly stable if limn_.,oo (Tnx, y) = 0 for every x E X and y E X'. Note that by the uniform boundedness principle every weakly stable operator

Ton a Banach space is power bounded and hence u(T) c {z: lzl::::; 1}. Moreover, the spectral conditions Pa (T) n r = 0 and Ra (T) n r = Pa (T') n r = 0 are necessary for weak stability.

(a) The left and right shifts are weakly stable on the spaces c0 (Z, X) and lP(Z, X) for any Banach space X and 1 < p < oo. Note that these operators are isometries and therefore not strongly stable.

Example 3.2.

(b) Consider H := l2 and the multiplication operator T given by

for some bounded sequence (an)~= 1 • Then Tis weakly stable if and only if ian I < 1 for every n E N. However, in this case T is automatically strongly stable. (c) The situation is different for a multiplication operator on L 2 (JR.) with respect to the Lebesgue measure J-L. Let T be defined as (T f) (s) := a( s) f (s) for some bounded measurable function a : JR. ----+ R Then T is strongly stable if and only if la(s)l < 1 for almost all s. However, the operator Tis weakly stable if and only if la(s)l ::::; 1 for almost all sand an(s)ds----+ 0 as n----+ oo for every interval [c, d] C R (Check weak convergence on characteristic functions and use the standard density argument.) This is the case for, e.g., a( s) = eias..,, ct,"( E JR.\ {0}. More examples will be given in Section 5.

t

We now present a simple condition implying weak stability and begin with the following definition, see, e.g., Furstenberg [92, p. 28].

Definition 3.3. A subsequence {nj }~ 1 of N is called relatively dense or syndetic if there exists a number£> 0 such that for every n E N the set {n, n intersects {nJ } ~ 1 .

+ 1, ... , n + £}

For example, kN + m for arbitrary natural numbers k and m is a relatively dense subsequence of N.

56

Chapter II. Stability of linear operators

Theorem 3.4. Let X be a Banach space and T E .C(X). Suppose that limj--+oo Tni = 0 weakly for some relatively dense subsequence {nj }~ 1 . Then T is weakly stable.

Proof. Define£:= supjEN(nJ+1- nj) which is finite by assumption and fix x EX and y E X'. For n E { nj, ... , nj + £} we have

(11.17) Note that Tn-nj x belongs to the finite set { x, Tx, . .. , Tfx}. By assumption we havelimj--+oo(z,T'niy) = Oforevery z EX, and (11.17) implieslimn--+oo(Tnx,y) = 0. 0 Remark 3.5. We will see later that one cannot replace relative density by density 1, see Section 4. We now state the following characterisation of weak convergence in terms of (strong) convergence of the Cesaro means of subsequences due to Lin [167], Jones, Kuftinec [136] and Akcoglu, Sucheston [2]. Theorem 3.6. Let X be a Banach space and T E .C(X). Consider the following assertions.

(i) Tnx converges weakly as n ____, oo for every x

E

X;

(ii) For every x EX and every increasing sequence {nj}~ 1 C .N with positive lower density, the limit

1 N

lim N

N-->oo

I: Tni x exists in norm for every x E X;

(II.18)

j=1

(iii) Property (II.18) holds for every x E X and every increasing sequence {nj}~ 1 C .N.

Then (i){:}(ii)-¢=(iii), and they all are equivalent provided X is a Hilbert space and T is a contraction. Here, the lower density of a sequence {nj }~ 1

· f #{j : nj d ·= 1·1mm _. n-->00

n

< n}

E

c .N is

defined by

[0 , 1] .

If the limit on the right-hand side exists, it is called density and denoted by d. Remark 3.7. Assertion (iii) in Theorem 3.6 is called the Blum-Hanson property and appears frequently in ergodic theory. For a survey on this property and some recent results and applications to some classical problems in operator theory (e.g., to the quasi-similarity problem) see Muller, Tomilov [192]. In particular, they showed that for power bounded operators on Hilbert spaces the Blum-Hanson property does not follow from weak convergence.

3. Weak stability

57

We also add a result of Muller [189] on possible decay of weak orbits. Theorem 3.8 (Miiller). LetT be an operator on a Banach space X with r(T) 2: 1 and {an}~=l be a positive sequence satisfying an ---+ 0. Then there exist x E X, y EX' and an increasing sequence {nj }~ 1 c N such that

Vj EN.

(II.19)

Surprisingly, inequality (II.19) does not hold for all n E N in general, i.e., the weak version of Theorem 2.8 is not true. For an example and many other phenomena see Muller [189] and [191, Section V.39]. However, for weakly stable operators one can choose nj := j, see Badea, Muller [12].

3.2

Contractions on Hilbert spaces

In this subsection we present classical decomposition theorems for contractions on Hilbert spaces having direct connection to weak stability. The first theorem is due to Sz.-Nagy, Foia§ [237], see also [238]. Theorem 3.9 (Sz.-Nagy, Foia§, 1960). Let T be a contraction on a Hilbert space H. Then His the orthogonal sum of two T- and T*-invariant subspaces H 1 and H2 such that

(a) H1 is the maximal subspace on which the restriction ofT is unitary; (b) the restrictions ofT and T* to H2 are weakly stable. We follow the proof given by Foguel [86].

Proof. Define

H1 := {x E H: IITnxll = IIT*nxll = llxll

for all n EN}.

We first prove that for every X E Hl and n EN one has T*nrnx = rnrmx = x. If X E Hl, then llxll 2 = (Tnx, rnx) = (T*nTnx, x) :::; IIT*nrnxllllxll :::; llxll 2 holds. Therefore, by the equality in the Cauchy-Schwarz inequality and positivity of llxll 2 we have T*nTnx = x. Analogously one shows rnr*nx = x. On the other hand, every x with these two properties belongs to H 1 . So we proved that

(II.20) is the maximal (closed) subspace on which Tis unitary. The T- and T* -invariance of H 1 follows from the definition of H 1 and the fact that T*T = TT* on H1. To prove (b) take x E H 2 := H{ Note that H2 is T- and T*-invariant since H 1 is. Suppose that Tnx does not converge weakly to zero as n ---+ oo. This means that there exists E > 0, y E H2 and a subsequence {nJ}~ 1 such that I(Tn1x,y)l2: f for every j EN. On the other hand, {Tn1x}~ 1 contains a weakly converging subsequence which we again denote by {Tn 1 x}~ 1 , and its limit by xo.

Chapter II. Stability of linear operators

58

Since H 2 is T-invariant and closed, xo belongs to H 2. To achieve a contradiction we show below that actually x 0 = 0. For a fixed kEN we have

IIT*kTkTnx- Tnxll2 = IIT*krk+nxll2 - 2(T*kTk+nx, rnx)

+ 11Tnxll2

::::; 11Tk+nxll 2 - 2IITk+nxll 2 + 11Tnxll 2 = 11Tnxll 2 -11Tk+nxll 2· The right-hand side converges to zero as n--+ oo since the sequence {IITnxll}~=l is monotone decreasing and therefore convergent. So IIT*kTkTnx- Tnxll --+ 0 as n--+ oo. We now return to the above subsequence {Tnix }~ 1 converging weakly to x 0 . Then T*kTkTni x --+ T*kTkx 0 weakly. On the other hand, by the considerations above, we have T*kTkTnix--+ xo weakly and therefore T*kTkxo = xo. One shows analogously that TkT*kx 0 = x 0 and hence x 0 E H 1 . Since H 1 n H 2 = {0}, we obtain x 0 = 0, the desired contradiction. Analogously, the powers of the restriction ofT* to H 2 converge weakly to

D

~-

Remark 3.10. The restriction ofT to the subspace H 2 in Theorem 3.9 is completely non-unitary (c.n.u. for short), i.e., there is no non-trivial subspace of H2 on which the restriction ofT becomes unitary. In other words, Theorem 3.9 states that every Hilbert space contraction can be decomposed into a unitary and a c.n. u. part, and the c.n.u. part is weakly stable. For a systematic study of completely non-unitary operators as well as an alternative proof of Theorem 3.9 using unitary dilation theory see the monograph of Sz.-Nagy and Foia§ [238].

On the other hand, we have the following decomposition into the weakly stable and the weakly unstable part due to Foguel [86]. We give a simplified proof. Theorem 3.11 (Foguel, 1963). LetT be a contraction on a Hilbert space H. Then

W := {x E H: lim (Tnx,x) = 0} n--->oo

coincides with {x E H: lim Tnx = 0 weakly}= {x E H: lim T*nx = 0 weakly} n---+oo

n-too

and is a closed T- and T* -invariant subspace of H, and the restriction of T to W _1_ is unitary. Proof. We first take x E W and show that Tnx--+ 0 weakly. By Theorem 3.9 we may assume that x E H 1 . TakeS:= lin{Tnx: n = 0, 1, 2 ... }. Since for ally E Sl_ we have (Tnx, y) = 0 for all n, it is enough to show that limn_, 00 (Tnx, y) = 0 for ally E S. For y := Tkx we obtain (Tnx, y) = (T*kTnx, x) = (Tn-kx, x)

--+

0

for k ::; n--+ oo,

3. Weak stability

59

where we used that the restriction of T to H 1 is unitary. From the density of lin{Tnx: n = 0,1,2, ... } inS, it follows that limn_, 00 (Tnx,y) = 0 for ev• ery y E S and therefore limn_,oo Tnx = 0 weakly. Analogously, one shows that limn->oo T*nx = 0 weakly. The converse implication, the closedness and the in• variance of W are clear. The last assertion of the theorem follows directly from Theorem 3.9. D Combining Theorem 3.9 and Theorem 3.11 we obtain the following decom• position. Theorem 3.12. LetT be a contraction on a Hilbert space H. Then H is the or• thogonal sum of three closed T- and T*-invariant subspaces H 1 , H 2 and H 3 such

that the corresponding restrictions T1, Tz and T3 satisfy 1. T1 is unitary and has no non-zero weakly stable orbit;

2. T2 is unitary and weakly stable; 3. T3 is completely non-unitary and weakly stable. The above theorem shows that a characterisation of weak stability for unitary operators is of special importance. For more aspects of this problem see Section

IV.l.

3.3 Characterisation via resolvent To close this section we give a resolvent characterisation of weak stability being a discrete analogue of a result for C0 -semigroups due to Chill, Tomilov [49], see Subsection III.4.3. Theorem 3.13. LetT be an operator on a Banach space X with r(T) :S: 1 and take

x EX andy EX'. Consider the following assertions.

(a)

(b)

(c)

lim (Tnx, y) = 0.

n->oo

Then (a):::}(b):::}(c). In particular, if (a) or (b) hold for all x E X andy E X', then T is weakly stable. Note that (a) means that the function z r---+ (R 2 (z, T)x, y) belongs to the Bergman space A1 ({z E C: 1 < lzl < 2}). (See Hedenmalm, Korenblum, Zhu [124] or Bergman [32] for basic information on Bergman spaces.)

60

Chapter II. Stability of linear operators

Proof. We first prove (a)=?(b). The function g : ][)) = {z E C :

lzl
00

= 0 in the weak

(iii*) There exists a subsequence {nj} ~ 1 C N with density 1 such that _lim Tni in the weak operator topology.

J-->00

= 0

Recall that the density of a sequence { nj} ~ 1 C N was defined after Theorem 3.6. The following elementary lemma (see, e.g., Petersen [212, p. 65] or Lemma III.5.2 below in the continuous case) will be needed in the proof of Theorem 4.1. Lemma 4.2 (Koopman-von Neumann, 1932). For a bounded sequence {an}~= 1 C

[0, oo) the following assertions are equivalent. 1 n (a) lim - ""'ak = 0. n-+oo

n L...J k=1

(b) There exists a subsequence {nj} ~ 1 of N with density 1 such that limj_, 00 ani = 0.

Chapter II. Stability of linear operators

62

Proof of Theorem 4.1. The implications (i')=}(i) and (ii)=}(vii) are trivial. (i)=}(ii) follows from the equivalence of weak compactness and weak sequen• tial compactness in Banach spaces (see Theorem I.1.1). The implication (vii)=}(i') is a consequence of Theorem !.1.15 and the con• struction in its proof. Therefore, we already proved the equivalences (i)¢:>(i')¢:>(ii) ¢::>(vii). (vi)=}(vii): Assume that there exists 0 f- x E X and 'P E [0, 27r) such that

Tx = ei'Px. Then we have IIR(rei'P,T)xii = (r -1)- 1 llxll and (vi) does not hold.

The converse implication follows from Theorem 2.8 and the fact that for every 0 :::; 'P < 27r the operator ei'PT has relatively weakly compact orbits as well. So (vi)¢:>( vii). (i')=}(iii): TakeS := {Tnx: n 2:: o{"(X) C .C(X) with the usual multipli• cation and the weak operator topology. It becomes a compact semitopological semigroup (for the definition and basic properties of compact semitopological semigroups see Subsection 1.1.4). By (i') we have 0 E S. Define the operator T : C(S) --. C(S) by

(Tf)(R)

:=

f(TR),

f

E

C(S), RES.

Note that T is a contraction on C(S). By Example I.l. 7 (c) the set {f (Tn ·) : n 2:: 0} is relatively weakly compact in C(S) for every f E C(S). It means that every set {Tn f : n 2:: 0} is relatively weakly compact, i.e., T has relatively weakly compact orbits. Denote by P the mean ergodic projection of T. We have Fix(T) = (1). Indeed, for f E Fix(T) one has f(Tn I) = f(I) for all n 2:: 0 and therefore f must be constant. Hence Pf is constant for every f E C(S). By definition of the ergodic projection

(Ff)(O) = lim - 1n-+oo

Thus we have

n +1

(F f)(R) = f(O) · 1,

t(Tk f)(O) = f(O). k=O f

E

C(S), R

E

S.

(II.22)

(II.23)

Take now x E X. By Theorem I.l.5 and its proof (see Dunford, Schwartz [63, p. 434]), the weak topology on the orbit {Tnx : n 2:: 0} is metrisable and coincides with the topology induced by some sequence {Yn}~=l C X'\ {0}. Consider fx E C (S) defined by

fx(R)

:=

L

nEN

}n 1\Rx, ~)I'

RES.

By (II.23) we obtain 0 = lim n-+oo n

1

n

-

1

L n

fx)(I) = lim fx(Tk). + 1 "'(Tk ~ n-->oo n + 1 k=O k=O

4. Almost weak stability

63

Lemma 4.2 applied to the sequence Ux(Tn 1)}~ 0 C JR.+ yields a subsequence (ni )fr= 1 of .N with density 1 such that lim fx(Tnj)

=

0.

J->00

By definition of fx and by the fact that the weak topology on the orbit is induced by {Yn}~=l we have that lim Tnjx

j-+oo

=0

weakly,

and (iii) is proved. (iii)==}(iv) follows directly from Lemma 4.2. (iv)==}(vii) is clear. (iv){::}(v): We note first that the set {Tn : n 2: 0} is bounded in £(X). Take x EX, y E X' and let r > 1. By (II.2) and the Parseval's equality we have

We obtain by the equivalence of Abel and Cesaro limits (see Lemma 1.2.6) that

1

211"

lim (r-1)

r-+1+

0

1

.

n

I(R(rez"',T)x,y)i 2 d'P=1f lim -_LI(Tkx,y)l 2 . (II.24) n-+oo n + 1 k=O

Note that for a bounded sequence { an}~=l C JR.+ with C := supnEN an we have

for every n E .N, where for the second part we used the Cauchy-Schwarz inequality. This together with (II.24) gives the equivalence of (iv) and (v). For the additional part of the assertion suppose X' to be separable. Then so is X, and we can take dense subsets {Xn =f. 0 : n E .N} ~ X and {Ym =f. 0 : m E .N} ~ X'. Consider the functions

fn,m : S -+JR.,

fn,m(R)

:=

1\RII~~II' 11~:11)1, n,m E .N,

which are continuous and uniformly bounded inn, mE .N. Define the function

f:

s-+

JR.,

J(R)

:=

L

n,mEN

1 2n+m fn,m(R).

Chapter II. Stability of linear operators

64

Then clearly f E C(S). Thus, as in the proof of the implication (i')=?(iii), i.e., using (11.22) we obtain 1 n lim "f(Tni) = 0. n-+oo n + 1 6

k=O

Hence, Lemma 4.2 applied to the bounded sequence {f(Tn)}~=O C JR.+ yields the existence of a subsequence {nj }~ 1 of N with density 1 such that limj-+oo f(Tnj) = 0. In particular, limj-+oo I(Tnj Xn, Ym) I = 0 for all n, m E N, which, together with the boundedness of {Tn}~= 1 , proves the implication (i')=?(iii*). The implications (iii*)=?(ii*)=?(ii') are straightforward, hence the proof is complete. 0 The above theorem shows that the property "no eigenvalues of T on the unit circle" implies properties like (iii) on the asymptotic behaviour of the orbits ofT. Moti• vated by this we introduce the following terminology.

Definition 4.3. We call an operator on a Banach space with relatively weakly compact orbits almost weakly stable if it satisfies condition (iii) in Theorem 4.1. Historical remark 4.4. Theorem 4.1 and especially the implication (vii)=?(iii) has a long history. It goes back to the origin of ergodic theory and von Neumann's spectral mixing theorem for flows, see Halmos [118], Mixing Theorem, p. 39. This has been generalised to operators on Banach spaces by many authors, see, e.g., Nagel [195], Jones, Lin [137, 138] and Krengel [154], pp. 108-110. Note that the equivalence (vii){:}(iv) for contractions on Hilbert spaces also follows from the so-called generalised Wiener theorem, see Goldstein [102]. For a continuous analogue of the above characterisation see Theorem III.5.1. Remark 4.5. We emphasise that the conditions appearing in Theorem 4.1 are of quite different nature. Conditions (i)-(iv), (ii*) and (iii*) describe the behaviour of the powers ofT, while conditions (v )-(vii) consider the spectrum and the resolvent ofT in a neighbourhood of the unit circle. Among them condition (vii) apparently is the simplest to verify. Note that one can also add the equivalent condition

L

n 1 sup -I(Tkx,y)l = 0 for every x EX, n-+oo yEX', IIYII:9 n + 1 k=O

(iv') lim

see Jones, Lin [137, 138].

Remark 4.6. The equivalence (i'){:}(v) is a weak analogue of the characterisation of strong stability given in Corollary 2.25. Remark 4. 7. Theorem 3.4leads to an interesting algebraic property of the sequence { nJ }~ 1 appearing in condition (iii). Indeed, if {nJ }~ 1 contains a relatively dense subsequence, then the operator Tis weakly stable. Reformulating this, we obtain

4. Almost weak stability

65

the following: Let T be almost weakly but not weakly stable. Then the sequence {nj }~ 1 in (iii) does not contain any relatively dense set (for example, it does not contain any equidistant sequence). Using Theorem 4.1 one can reformulate the Jacobs-Glicksberg-de Leeuw decomposition (see Theorem !.1.15) as follows. Theorem 4.8 (Jacobs-Glicksberg-de Leeuw decomposition, extended version). Let

X be a Banach space and letT E L:(X) have relatively weakly compact orbits. Then X= Xr EB X 8 , where Xr :=lin{x EX: Tx =/X for some rEf}, Xs := { x E X : lim Tnj x = 0 weakly for some subsequence {nj} ~ 1 J--+00

with density

1}.

One also can formulate Theorem 4.1 for single orbits. This is the following result partially due to Jan van Neerven (oral communication) in the continuous case. Corollary 4.9. Let T be an operator on a Banach space X and x E X. Assume that the orbit {Tnx : n = 0, 1, 2, ... } is relatively weakly compact in X and the

restriction ofT to lin{Tnx : n = 0, 1, 2, ... } is power bounded. Then there is a holomorphic continuation of the function R(·, T)x to {>. : 1>-1 > 1} denoted by Rx (·) and the following assertions are equivalent.

(ii) There exists a subsequence {nj} ~ 1 C N such that limj--+oo Tnj x = 0 weakly. (iii) There exists a subsequence {n1 }~ 1 c N with density 1 such that lim1__, 00 Tnj x = 0 weakly.

1 (iv) lim - n--+oo n + 1

L I(Tkx, y)l = 0 for ally EX'. n

1

k=O

21f

(v) lim (r-1) r-->1+

0

I(Rx(rei'P),y)l 2 d'{J=OforallyEX'.

(vi) lim (r -1)Rx(reirp) = 0 for all 0 ~ 'P r--+1+

< 27r.

(vii) The restriction ofT on lin{Tnx: n E NU{O}} has no unimodular eigenvalue.

Proof. For the first part of the theorem we just define Rx(>.) :=

00 rnx >,n+ 1

L

n=O

whenever 1>-1 > 1.

Chapter II. Stability of linear operators

66 This implies the representation

Denote now by Z the closed linear span of the orbit {Tnx : n = 0, 1, 2, ... }. Then Z is aT-invariant closed subspace of X and we can restrict T to it. The restriction, which we denote by Tz, has relatively weakly compact orbits by Lemma I.l.6. The equivalence of the assertions follows from the canonical decomposition X' = Z' EB Z 0 with Z 0 := {y E X' : (z, y) = 0 for all z E Z} and Theorem 4.1 applied to the restricted operator. 0

4.2

Concrete example

As we will see from the abstract examples in the next section, almost weak stability does not imply weak stability. In this subsection we present a concrete example of a (positive) operator being almost weakly but not weakly stable. (For definitions and basic theory of positive operators we refer to Schaefer [227].) Example 4.10. Consider the operator To(1) from Example III.5.9 below being a positive operator on the Banach lattice C(O) for some 0 c C. The relative weak compactness of the orbits follows from the relative weak compactness of the semigroup To(·). The almost weak stability ofT0 (1) is a consequence of Pa(T0 (1) )n r = 0 and Theorem 4.1. Further, since the semigroup To(-) is not weakly stable and N is a relatively dense set in JR+, it follows from Theorem III.4.4 that the operator To(1) is not weakly stable.

So we proved the following result. Theorem 4.11. There is a locally compact space 0 and a positive contraction T on

Co (0) which is almost weakly but not weakly stable. Note that for positive operators on C(K) and L 1-spaces, almost weak sta• bility does imply weak stability and even a stronger stability property, see Chill, Tomilov [50] as well as Groh, Neubrander [112, Theorem. 3.2] for the continuous case. Theorem 4.12. For a power bounded, positive, and mean ergodic operator T on a

Banach lattice X, the following assertions hold. (i) If X~ L 1 (0,f.l), then P(J(T')

nr

=

0 is equivalent to

(ii) If X ~ C(K), K compact, then P(J(T') exponential stability ofT.

strong stability ofT.

n r = 0 is equivalent to

uniform

5. Category theorems

67

5 Category theorems By proving category results, we obtain abstract examples of almost weakly but not weakly stable operators and show that the difference between these two concepts (at least for operators on Hilbert spaces) is dramatic. More precisely, we show that a "typical" (in the sense of Baire) contraction as well as a "typical" isometric or unitary operator on a separable Hilbert space is almost weakly but not weakly stable. This gives an operator-theoretic analogue to the classical theorems of Hal• mos and Rohlin from ergodic theory stating that a "typical" measure preserving transformation is weakly but not strongly mixing, see Halmos [118, pp. 77-80] or the original papers by Halmos [116] and Rohlin [221], see also Section IV.2. We follow in this section Eisner, Sen~ny [69] and assume the underlying Hilbert space H to be separable and infinite-dimensional.

5.1

Unitary operators

Denote the set of all unitary operators on H by U. The following density result for periodic operators is a first step in our construction. Proposition 5.1. For every n E N the set of all periodic unitary operators with period greater than n is dense in U endowed with the operator norm topology. Proof. Take U E U, N E N and E > 0. By the spectral theorem H is isomorphic to L 2 (0, J-t) for some finite measure J-t on a loc~lly compact space nand U is unitarily equivalent to the multiplication operator U with

(U f)(w) = 'P(w)f(w) for some measurable 'P : n -+ r. We approximate the operator

rN

:= { e21ri· ~

:

for almost all wEn,

U as follows.

Consider the set

p, q E N relatively prime, q > N}

which is dense in r. Take a finite set {oj}j=1 c rN with 0:1 = O:n such that arg(aj_ 1) < arg(aj) and lai- O:j-11 < E hold for all2::; j::; n. Define

and denote by P the multiplication operator with 1/J. The operator with period greater than N and satisfies IIU- Fll =sup I'P(w)- 1/J(w)l::;

P is

periodic

E,

wE!l

hence the proposition is proved. For the second step we need the following lemma.

D

68

Chapter II. Stability of linear operators

Lemma 5.2. Let H be a separable infinite-dimensional Hilbert space. Then there

exists a sequence {Tn}~=l of almost weakly stable unitary operators satisfying limn-+oo IITn - Ill = 0. Proof. It suffices to prove the result for H = L2 (JR). Taken EN and define Tn on L 2 (1R) by -

iq(s)

(Tnf)(s) :=e-n f(s), s E lR, f E L2 (1R), where q : lR-* [0, 1] is strictly monotone. Then all Tn are almost weakly stable by Theorem 4.1, and they approximate I since IITn- Ill =sup le-n -

iq(s)

sEIR

- 11 :::;

i

len: -

11 -* 0,

n-*

00.

0

The lemma is proved.

We now introduce the strong* (operator) topology which is induced by the family of seminorms Px(T) := JIITxll 2 + IIT*xll 2 , x E H. We note that conver• gence in this topology corresponds to strong convergence of operators and their adjoints. For properties and further information on this topology we refer to Take• saki [239, p. 68]. In the following we consider the space U of all unitary operators on H en• dowed with this strong* operator topology. Note that U is a complete metric space with respect to the metric given by

and {x1 }~ 1 some dense subset of H \ {0}. Further we denote by Su the set of all weakly stable unitary operators on H and by Wu the set of all almost weakly stable unitary operators on H. We now show the following density property for Wu. Proposition 5.3. The set Wu of all almost weakly stable unitary operators is dense

inU. Proof. By Proposition 5.1 it is enough to approximate periodic unitary operators by almost weakly stable unitary operators. Let U be a periodic unitary operator and let N be its period. Take E > 0, n EN and x 1 , ... ,xn E H \ {0}. We have to find an almost weakly stable unitary operator T with IIUxj- Tx1ll < E and IIU*xj - T*xj II < E for all j = 1, ... , n. Since U

N

=I we have a(U) c

{

21ri

21r(N -l)i }

1, er:r, ... , e--N-

, and the orthogonal

decomposition 27ri

21r(N -l)i

H = ker(I- U) EB ker(el\f I- U) EB ... EB ker(e_N_I- U)

(11.25)

5. Category theorems

69

holds. Assume first that XI, ... , Xn are orthogonal eigenvectors of U. In order to use Lemma 5.2 we first construct a periodic unitary operator S satisfying UXj = Sxj for all j = 1, ... , n and having infinite-dimensional eigenspaces only. For this purpose define the n-dimensional U- and U*- invariant subspace Ho := lin{xj}J=I and the operator So on Ho as the restriction of U to H0 . Decompose Has an orthogonal sum 00

H = EBHk with dimHk = dimH0 for all kEN. k=O Denote by Pk an isomorphism from Hk to H0 for every k. Define now Sk ·• P;;Iupk on each Hk as a copy of UIHo and considerS:= ffi~o Sk on H. The operator S is unitary and periodic with period being a divisor of N. So a decomposition analogous to (II.25) is valid for S. Moreover, Uxj = Sxj and U*xj = S*xj hold for all j = 1, ... , nand the eigenspaces of S are infinite dimen• sional. Denote by Fj the eigenspace of S containing Xj and by Aj the corresponding eigenvalue. By Lemma 5.2, for every j = 1, ... , n there exists an almost weakly stable unitary operator Tj on Fj satisfying IITj- SIFJ II = IITj- >..jill < c. Finally, we define the desired operator T as Tj on Fj for every j = 1 ... , n and extend it linearly to H. Let now XI, ... , Xn E H be arbitrary and take an orthonormal basis of eigen• values {yk}f:I· Then there exists K E N such that Xj = ~~=I ajkYk + Oj with lloj II < ~ for every j = 1, ... , n. By the arguments above applied to YI, ... , YK and there is an almost weakly stable operator T with IIUYk - Tykll < forM:= maxk=I, ... ,K,j=l, ... ,n laikl and every k = 1, ... ,K. IIU*yk-T*ykll < Therefore we obtain

4;M

4;M

K

IIUxj- Txjll :S

L

k=I

laikiiiUYk- Tykll

+ 2llojll < c

for every j = 1, ... ,n. Analogously, IIU*xj-T*xjll < c holds for every j = 1, ... ,n, 0 and the proposition is proved. We can now prove the following category theorem for weakly and almost weakly stable unitary operators. To do so we extend the argument used by Halmos and Rohlin (see, e.g., [118, pp. 77-80]) for measure preserving transformations in ergodic theory.

Theorem 5.4. The set Su of weakly stable unitary operators is of first category and the set Wu of almost weakly stable unitary operators is residual in U. Recall that a subset of a Baire space is called residual if its complement is of category. first

Chapter II. Stability of linear operators

70

Proof. We first prove that Sis of first category in U. Fix x E H with consider the closed sets

llxll = 1 and

Let U E U be weakly stable. Then there exists n EN such that U E Mk for all k 2 n, i.e., U E nk?.nMk. So we obtain 00

(II.26) where Nn := nk?.nMk. Since the sets Nn are closed, it remains to show that U\Nn is dense for every n. Fix n E N and let U be a periodic unitary operator. Then U ~ Mk for some k 2 n and therefore U ~ Nn. Since by Proposition 5.1 the periodic unitary operators are dense in U, S is of first category. To show that Wu is residual we take a dense subset D = {xj}~ 1 of Hand define the open sets

Then the sets WJk := We show that

U~=l Wjkn

are also open.

n 00

Wu =

(11.27)

Wjk

j,k=l

holds. The inclusion "c" follows from the definition of almost weak stability. To prove the converse inclusion we take U ~ Wu. Then there exists x E H with . 1 llxll = 1 and 'P E IR such that Ux = e''Px. Take now Xj E D with llxJ - xll :::; 4· Then

I (Unxj, Xj) I= I(Un(x- Xj), X- Xj) + (Unx, x)- (Unx, X- Xj) - (Un(x- Xj ), x) I

2 1-

2

1

llx- Xj I - 2llx- Xj I > 3

holds for every n E N. So U ~ WJ3 which implies U ~ n_rk=l WJk, and therefore (II.27) holds. Moreover, all Wjk are dense by Proposition 5.3. Hence Wu is residual as a countable intersection of open dense sets. D

5. Category theorems

71

5.2 Isometries We now consider the space I of all isometries on H, a separable and infinite• dimensional Hilbert space, endowed with the strong operator topology and prove analogous category results as above. Note that I is a complete metric space with respect to the metric given by the formula d(T, S) :=

f

j=l

IITxj - Sxj 2J llxj I

II forT, S

E

I,

where {xJ}~ 1 is a fixed dense subset of H \ {0}. Further we denote by Sr the set of all weakly stable isometries on H and by Wr the set all almost weakly stable isometries on H. The results in this subsection are based on the following classical theorem on isometries on Hilbert spaces, see [238, Theorem 1.1].

Theorem 5.5 (Wold decomposition). Let V be an isometry on a Hilbert space H. Then H can be decomposed into an orthogonal sum H = Ho EB H 1 of V -invariant subspaces such that the restriction of V on Ho is unitary and the restriction of V on H 1 is a unilateral shift, i.e., there exists a subspace Y c H 1 with vny l. vmy for all n of m, n, mEN, such that H1 = EB~=l vny holds. We need the following easy lemma, see also Peller [211].

Lemma 5.6. Let Y be a Hilbert space and let R be the right shift on H := l2 (N, Y). Then there exists a sequence {Tn}~=l of periodic unitary operators on H converg• ing strongly to R. Proof. We define the operators Tn by

Every Tn is unitary and has period n. Moreover, for an arbitrary x = (x1, x2, ... ) E H we have 00 IITnX- Rxll 2 = llxn 11 2 +

2::: llxk+l

k=n

- Xk 11 2 n-=:;: 0,

and the lemma is proved.

D

As a first application of the Wold decomposition we obtain the density of the periodic operators in I. (Note that periodic isometries are unitary.)

Proposition 5.7. The set of all periodic isometries is dense in I. Proof. Let V be an isometry on H. Then by Theorem 5.5 the orthogonal decompo• sition H = H0 EB H1 holds, where the restriction Vo on Ho is unitary and the space H 1 is unitarily equivalent to l2(N, Y). The restriction V1 of V on H1 corresponds (by this equivalence) to the right shift operator on l 2 (N, Y). By Proposition 5.1 and Lemma 5.6 we can approximate both operators Vo and V1 by unitary periodic ones and the assertion follows. D

Chapter II. Stability of linear operators

72

We further obtain the density of the set of all almost weakly stable operators in I. Proposition 5.8. The set

W:r of almost weakly stable isometries is dense in I.

Proof. Let V be an isometry on H and let Vo and V1 be the corresponding re• strictions of V to the orthogonal subspaces H 0 and H 1 from Theorem 5.5. By Lemma 5.6 the operator V1 can be approximated by unitary operators on H1. The assertion now follows from Proposition 5.3. 0 Using the same idea as in the proof of Theorem 5.4 one obtains, using Propo• sitions 5.7 and 5.8, the following category result for weakly and almost weakly stable isometries. Theorem 5.9. The set Sr of all weakly stable isometries is of first category and

the set Wr of all almost weakly stable isometries is residual in I.

5.3

Contractions

The above category results are now extended to the case of contractive operators. Let the Hilbert space H be as before and denote by C the set of all contrac• tions on H endowed with the weak operator topology. Note that C is a complete metric space with respect to the metric given by the formula

where {xj}~ 1 is a fixed dense subset of H with each Xj i- 0. By Takesaki [239, p. 99], the set of all unitary operators is dense in C (see also Peller [211] for a much stronger assertion). Combining this with Propositions 5.1 and 5.3 we have the following fact. Proposition 5.10. The set of all periodic unitary operators and the set of all almost

weakly stable unitary operators are both dense in C. The following elementary property is a key for the further results (cf. Halmos [121, p. 14]).

Lemma 5.11. Let {Tn}~=l be a sequence of linear operators on a Hilbert space H converging weakly to a linear operatorS. If IITnxll :::; IISxll for every x E H, then lim Tn = S strongly. n-->oo

Proof. For each x E H we have IITnx-

Sxll 2 = (Tnx- Sx, Tnx- Sx) = IISxll 2 + 11Tnxll 2 -

2Re (Tnx, Sx) :::; 2 (Sx, Sx)- 2Re (Tnx, Sx) = 2Re ((S- Tn)x, Sx) ------r 0, n-->oo

and the lemma is proved.

0

5. Category theorems

73

We now state the category result for contractions, but note that its proof differs from the corresponding proofs in the two previous subsections. Theorem 5.12. The set Se of all weakly stable contractions is of first category and

the set We of all almost weakly stable contractions is residual in C.

Proof. To prove the first statement we fix x E X, llxll = 1, and define as before the sets

Nn := { T E C : I(Tkx, x) I : : ;

~ for all k ?_ n

} .

LetT E C be weakly stable. Then there exists n E N such that T E Nn, and we obtain 00

SeC

UNn.

(II.28)

n=l

It remains to show that the sets Nn are nowhere dense. Fix n EN and let U be a periodic unitary operator. We show that U does not belong to the closure of Nn. Assume the opposite, i.e., that there exists a sequence {Tk}kEN c Nn satisfying limk-->oo Tk = U weakly. Then, by Lemma 5.11, limk-->oo Tk = U strongly and therefore U E Nn by the definition of Nn. This contradicts the periodicity of U. By the density of the set of unitary periodic operators inC we obtain that Nn is nowhere dense and therefore Se is of first category. To show the residuality of We we again take a dense subset D = {xj}~ 1 of Hand define

As in the proof of Theorem 5.4 the equality

n 00

We=

WJk

(II.29)

j,k=l

holds. Fix j, k E N. We have to show that the complement WJk of Wjk which coincides with {TEC: I(Tnxj,XJ)I ?_

~ forallnEN}

is nowhere dense. Let U be a unitary almost weakly stable operator. Assume that there exists a sequence {Tm}~=l C WJk satisfying limm-->oo Tm = U weakly. Then, by Lemma 5.11, limm-->oo Tm = U strongly and therefore U E WJk· This contradicts the almost weak stability of U. Therefore the set of all unitary almost weakly stable operators does not intersect the closure of WJk· By Proposition 5.10 0 all the sets WJk are nowhere dense and therefore We is residual.

74

Chapter II. Stability of linear operators

Remark 5.13. As a consequence of Theorem 5.12 we see that the set of all strongly stable operators as well as the set of all operators T satisfying r(T) < 1 are also of first category in C. Open question 5.14. Do the above category theorems hold on reflexive Banach spaces?

On non-reflexive Banach spaces these results are not true in general. Indeed, every almost weakly stable contraction on the space l 1 is automatically weakly and even strongly stable by Schur's lemma, see, e.g., Conway [52, Prop. V.5.2], and Lemma 2.4. For related category results see e.g. Iwanik [82], Bartoszek [16], Bartoszek, Kuna [18].

Remark 5.15. In Section IV.3 we prove a stronger version of the above results.

6 Stability via Lyapunov's equation Our spectral and resolvent criteria for stability and power boundedness of an operator T used information on R(>.., T) in many points>.. E p(T), i.e., one had to solve the equation (>..I- T)x = y for ally and infinitely many>.. (and then even had to estimate the solutions). This is, in practical situations, a quite difficult task. In a Hilbert space H, one can reduce this problem to the solution of a single equation in a new, but higher dimensional space. The idea is to use the imple• mented operators on and the order structure of C(H), see Section I.4. Recall that implemented operators are positive operators on C(H) defined as TS := T* ST for some fixed operator T E C(H).

6.1

Uniform exponential and strong stability

We start by characterising uniform exponential stability. Theorem 6.1. Let T be a bounded operator on a Hilbert space H and T its imple•

mented operator on C(H). Then the following assertions are equivalent.

(i) T is uniformly exponentially stable on H. (ii) T is uniformly exponentially stable on C(H).

(iii) 1 E p(T) and 0:::; R(1, T). (iv) There exists 0 :::; Q E C(H) such that

(T- I)Q = T*QT- Q =-I.

(II.30)

6. Stability via Lyapunov's equation

75

(v) There exists 0::; Q E £(H) such that T*QT- Q::; -I. In this case, Q in (iv) is unique and given by Q

= R(l, T)I = L~=O T*nrn.

Equation (II.30) is called Lyapunov equation.

Proof. (i){:}(ii) follows from Lemma !.4.6, (ii){:}(iii) by Theorem !.4.4 and (iii) implies (iv) by taking Q := R(l, T)I ~ 0. Since (iv):::}(v) is trivial, we now assume (v) and prove (iii). Take 0 :S: Son Hand observe that S :S: IISII· I. Since (I- T)Q = I we obtain n

n

n

k=O

k=O

k=O

This shows that the sequence {l::~=O TkS}~=O is monotonically increasing and bounded, hence the series Tk S converges weakly for every 0 ::; S and hence for every S E £(H) by Lemma !.4.1. So 1 E p(T) and R(1, T) = L~o Tk (for the strong operator topology) implying R(1, T) ~ 0, and (c) is proved. 0

I:%"=o

We now turn our attention to strong stability. Theorem 6.2. LetT E £(H) and T E £(£(H)) be as above. Then the following

assertions are equivalent.

(i) T is strongly stable. (ii) limn_, 00 (TnSx,x) (iii) For every a

= 0 for every S

E

> 0 there exist (unique)

£(H) and x E H. 0 :S: Qa, Qa E £(H) satisfying

such that lim a(Qax, x)

a--+0+

The operators Qa and

= 0 and sup a(Qax, x) < oo for all x

Qa are

a>O

E

H.

(II.31)

obtained as

Here, T* denotes the operator implemented by T*.

Proof. The implication (ii):::}(i) follows from the relation 11Txll 2 = (Tix, x). To prove the converse takeS E l(H)sa and observe -cl :S: S :S: ci for c := IISII· This implies I(TnSx, x)l :S: ci(Tn Ix, x)l = ci1Tnxll 2 proving (i):::}(ii). Assume that (i) or (iii) hold. Then Theorem 6.1 applied to e-ar and its implemented operator Ta = e- 2aT implies that Qa := R(1, Ta)I = e2aR(e 2a, T)I

Chapter II. Stability of linear operators

76

is the unique solution of the first rescaled Lyapunov equation in (iii) for every a> 0. Moreover, 00

00

n=O

n=O

Analogously, Qa := e2aR(e 2 a, T*)I is the unique solution of the second rescaled 2ani1Tnxll 2 . Lyapunov equation in (iii) and satisfies (Qax, x) = We now see that (II.31) is equivalent to the property

I::=o e-

L

a-->0+ n=O

L

00

00

lim a

e- 2ani1Tnxll 2 = 0 and

sup a

a>O n=O

e- 2aniiT*nxll 2

< oo.

By Lemma 1.2.6, this is equivalent to

n->oo

1

L 11Tkxll n + 1 k=O

lim -

n

2

= 0

and

1 L IIT*kxll n>O n + k=O

sup -

1

n

2

< oo

for every x E H which is equivalent to strong stability ofT by Proposition 2.7, D and therefore (i){:}(iii).

Remark 6.3. Note that one can again replace "= -I" by

"~

-I" in the rescaled

Lyapunov equations in (iii).

6.2 Power boundedness One can also characterise power boundedness of operators on Hilbert spaces via analogous Lyapunov equations.

Theorem 6.4. ForTE £(H) and T E £(£(H)) as before, the following assertions are equivalent.

(i) T is power bounded. (ii) T is power bounded.

(iii) For every a > 0 there exist (unique) 0 ~ Qa, Qa

E

£(H) satisfying

e- 2aT*QaT- Qa =-I and e- 2aTQaT*- Qa =-I such that supa(Qax,x)

a>O

< oo and

In this case, one has Qa

supa(Qax,x)

a>O

= e2aR(e 2a, T)I

and Qa

< oo for all x E H.

(II.32)

= e2aR(e 2a, T*)I.

The implication (i){:}(ii) follows directly from Lemma 1.4.6, while the proof of (i){:}(iii) is analogous to the proof of Theorem 6.2 using Proposition 1.7.

6. Stability via Lyapunov's equation

77

Final remark Again, weak stability is much more delicate to treat than strong stability or power boundedness. The main problem occurs when Tis a unitary operator. (Note that by Theorem 3.9, the unitary part of a contraction is decisive for weak stability.) In this case, the implemented operator T satisfies T I = I, and therefore weak stability cannot be described in terms of {Tn I}~=O or R(·, T)I, respectively, as we could do in Theorems 6.1 and 6.2. Open question 6.5. Is it possible to characterise weak stability of operators on Hilbert spaces via some kind of Lyapunov equation?

Chapter III

Stability of Co-semigroups In this chapter we discuss boundedness, polynomial boundedness, exponential, strong, weak and almost weak stability of C0 -semigroups (T(t))t~o on Banach spaces. The goal is to characterise these properties without using the semigroup explicitly. The optimal case is to have a characterisation in terms of the spectrum of the generator A. However, such results are possible only in very special cases. As it turns out, a good substitute is the behaviour of the resolvent R().., A) in a neighbourhood of iR In our presentation we emphasise the similarities and differences to the dis• crete case treated in Chapter II.

1 Boundedness In this section we consider boundedness and the related notion of polynomial boundedness for Co-semigroups. While boundedness of Co-semigroups is difficult to check if the semigroup is not contractive, the weaker notion of polynomial boundedness is easier to characterise.

1.1 Preliminaries We start with bounded C0 -semigroups and their elementary properties.

Definition 1.1. A C0 -semigroup T(·) on a Banach space X is called bounded if SUPt~O

IIT(t)ll < 00.

Remark 1.2. Every bounded semigroup T(·) satisfies wo(T) ::; 0 and hence a( A) C {z: Re(z)::; 0}. However, the spectral condition a(A) C {z: Re(z)::; 0} does not imply boundedness of the semigroup, as can be seen already from the matrix

80

Chapter III. Stability of C 0 -semigroups

semigroup T(·) given by T(t) =

D

(~

on C 2 . In Subsection 1.3 we give more

sophisticated examples and a general description of possible growth. Remark 1.3. It is interesting that, analogous to power bounded operators (see Subsection 1.1), more information on the spectrum of the generator of a bounded Co-semigroup is known if X is separable. For example, Jamison [135] proved that if a Co-semigroup on a separable Banach space is bounded, then the point spectrum of its generator on the imaginary axis has to be at most countable. For more results in this direction see Ransford [218], Ransford, Roginskaya [219] and Badea, Grivaux [14]. We will see later that countability of the spectrum of the generator on the imaginary axis plays an important role for strong stability of the semigroup, see Subsection 3.3.

The following simple lemma is useful to understand boundedness. Lemma 1.4. Let TC) be a bounded semigroup on a Banach space X. Then there exists an equivalent norm on X such that T(·) becomes a contraction semigroup. Proof. Take

llxll1 := supt;:::o IIT(t)xll for every x EX.

0

Remark 1.5. Packel [207] showed that not all bounded C0 -semigroups on Hilbert spaces are similar to a contraction semigroup for a Hilbert space norm. His example was a modification of the corresponding examples of Foguel [87] and Halmos [120] in the discrete case. However, Sz.-Nagy [236] proved that every bounded C0 -group is similar to a unitary one. We also mention Vii and Yao [253] who proved that every bounded uniformly continuous quasi-compact Co-semigroup on a Hilbert space is similar to a contraction semigroup.

Zwart [263] and Guo, Zwart [113, Thm. 8.2] showed that some kind of Cesaro boundedness implies boundedness. See also van Casteren [45] for the case of bounded C0 -groups on Hilbert spaces. Theorem 1.6. For a C 0 -semigroup T(·) on a Banach space X the following asser• tions are equivalent.

(a) T(·) is bounded. (b) For all x EX andy EX' we have

11t IIT(s)xll 11t IIT'(s)yll

supt;:::o t o

2 ds

< oo,

sup-

2 ds

< oo.

t20

t

0

1. Roundedness

81

Proof. (a)::::}(b) is clear. (b)::::}(a). Take x E X andy E X'. Then we have by the Cauchy-Schwarz inequality I(T(t)x, y)l

11t

=-

t

~

0

I(T(t)x, y)l ds

11t

=-

t

t

0

t

! 2

(lfo11T(s)xll ds)

I(T(t- s)x, T'(s)y)l ds

2

! 2

(lfo11T'(s)yll ds)

2 ,

and hence every weak orbit { (T(t)x, y) : t 2:: 0} is bounded by assumption. SoT(·) is bounded by the uniform boundedness principle. D Remark 1.7. The second part of condition (b) in the above theorem cannot be omitted, i.e., absolute Cesaro-boundedness is not equivalent to boundedness. Van Casteren [45] gave an example of an unbounded C0 -group on a Hilbert space satisfying the first part of condition (b).

Theorem 1.6 can also be formulated for single orbits. Proposition 1.8. For a C0 -semigroup T(·) on a Banach space X, x EX andy EX' the weak orbit {(T(t)x, y) : t 2:: 0} is bounded if and only if there exist p, q > 1 with .!p + .!q = 1 such that

11t 11t

sup-

t

0

supt~O t

0

t~O

IIT(s)xiiP ds < oo, IIT'(s)yllq ds

< oo.

However, since the semigroup is, in most cases, not known explicitly, it is important to find characterisations not involving it directly.

1.2 Representation as shift semigroups It is well-known that every bounded semigroup on a Banach space is similar to the left shift semigroup on a space of continuous functions. Theorem 1.9. LetT(·) be a contractive Co-semigroup on a Banach space X. If T(-) is strongly stable, then it is isometrically equivalent to the left shift semigroup on a closed subspace of BUC(JR+,X).

By BUC(JR+, X) we denote the space of all bounded, uniformly continuous functions from JR+ to X.

Proof. Define the operator J:

X~

BUC(JR+, X) by

(Jx)(s) := T(s)x,

s 2:: 0, x EX.

82

Chapter III. Stability of C0 -semigroups

This operator identifies a vector x with its orbit under the semigroup. (Note that every function s I--) T( s )x is uniformly continuous by the assumption on T( ·).) By contractivity ofT(·) we have llxll = maxs?:O IIT(s)xll = IIJxll and hence J is an isometry. Moreover,

JT(t)x = T(t + ·)x,

t ~ 0, x EX,

i.e., T(·) corresponds to the left shift semigroup on rg J which is a closed shift• D invariant subspace of C0 (JR.+, X). By the renorming procedure from Lemma 1.4 we obtain the following general form of bounded semigroups. Corollary 1.10. LetT(·) be a strongly stable semigroup on a Banach space X. Then T(·) is isomorphic to the left shift semigroup on a closed subspace of BUC(JR.+, X 1), where X 1 = X endowed with an equivalent norm.

1.3

Characterisation via resolvent

In this subsection we characterise boundedness for C0 -semigroups by the first or second power of the resolvent of its generator. Theorem 1.11 (Gomilko [104], Shi and Feng [231]). Let A be a densely defined operator on a Banach space X satisfying s0 (A) :::; 0. Consider the following asser•

tions. (a) For every x EX andy EX', sup a foo a>O

sup a a>O

(b) sup af a>O

00

-oo

!

-oo 00

-oo

IIR(a +is, A)xll 2 ds < oo, IIR(a +is, A')yll 2 ds < oo;

I(R 2 (a+is,A)x,y)lds < oo for all x EX, y EX';

(c) A generates a bounded Co-semigroup on X. Then (a)=?(b)=;.(c). Moreover, if X is a Hilbert space, then (a){:}(b){:}(c).

i:

i:

Proof. Assume (a). By the Cauchy-Schwarz inequality we have a

I(R 2 (a+is,A)x,y)lds:::; a

which proves (b).

IIR(a+is,A)xiiiiR(a+is,A')yllds

1. Roundedness

83

Consider now the implication (b)=>(c). We will construct the semigroup ex• plicitly using the idea from Shi, Feng [231]. We first prove that (b) implies so(A):::; 0. Since fzR(z,A) = -R2 (z,A), we have for all a> 0, x EX andy EX' that

(R(a +is, A)x, y) = (R(a, A)x, y) - i los (R(a +iT, A) 2 x, y) dr.

(III.l)

By the absolute convergence of the integral on the right-hand side we obtain that lim 8 ..... 00 (R(a+is,A)x,y) = 0. From (III.l) and (b) it follows that

.

M

IIR(a + zs, A) II :::; - , a

hence s 0 (A) :::; 0 holds. Define now T(O) =I and

T(t)x = 1-

1 00

27rt _ 00

2 xds . e(a+~s)tR(a+is,A)

for all x E X and t > 0. By the assumption the operators T(·) are well-defined and form a semigroup by Theorem 1.3.9. Let us estimate IIT(t)ll· By (b) and the uniform boundedness principle we have

eat

1oo

I(R(a+is,A) 2 x,y)ids I(T(t)x,y)l:::;27rt _ 00 Meat

::::: 21rta llxiiiiYII· Taking a:=

r

1

we obtain forK:= ~: the desired estimate IIT(t)ll ::::: K \:ft 2: 0.

(111.2)

Finally, this and Theorem 1.3.9 imply the strong continuity ofT(·). For the implication (c)=>(a), let T(·) be a Co-semigroup on a Hilbert space H with generator A satisfying K := supt~o IIT(t)ll < oo. By Parseval's inequality and integration by parts we have

and analogously

for every x, y E H, and the theorem is proved.

D

84

Chapter III. Stability of C 0 -semigroups

Remarks 1.12. 1) Gomilko first proved the non-trivial implication (b)::::}(c) using the Hille-Yosida theorem. Then Shi and Feng gave an alternative proof using an explicit construction of the semigroup by formula (I.13) in Subsection I.3.2. The direction (c)::::}(a) for Hilbert spaces follows easily from Plancherel's theorem. Note that van Casteren [45] presented an analogous characterisation for bounded C0 -groups on Hilbert spaces much earlier. He also showed that the second part of condition (a) cannot be omitted.

2) One can replace the £ 2 -norms in condition (a) by the £P-norm in the first inequality and Lq-norm in the second one for p, q > 1 with .!p + .!q = 1, possibly depending on x and y. However, for the converse implication, if X is a Hilbert space, one needs p = q = 2. A direct consequence of the construction given in the proof of Theorem 1.11 is the following result for single orbits. Proposition 1.13. Let X be Banach space and A generate a Co-semigroup T(·)

with s0 (A) :S 0, x EX andy EX'. Consider the following assertions.

(a) For some p,q > 1 with~+ limsup aj

00

limsup aj

00

a-+0+

a-+0+

(b) limsup aj a-+0+

00

-oo

i = 1,

-oo

-oo

IIR(a+is,A)xiiPds < oo, IIR(a+is,A')yllqds < oo;

I(R 2 (a+is,A)x,y)lds

< oo;

(c) { (T(t)x, y) : t 2:: 0} is bounded. Then (a)::::}(b)::::}(c). Moreover, if X is a Hilbert space, then (a){:}(b){:}(c) for p = q = 2.

In particular, conditions (a) and (b) holding for all x E X and y E X' both imply boundedness ofT(·) and are equivalent to it when X is a Hilbert space and

p = q = 2.

Note that conditions (a) and (b) in Theorem 1.11 are not necessary for boundedness since Corollary 2.19 below shows that the integrals in (a) and (b) can diverge in general. But even if all these integrals converge, condition (a) is still not necessary for boundedness even for isometric groups with bounded generators. This can be seen from the following example.

Example 1.14. On X := C[0,1] consider the bounded operator A defined by Af(s) := isf(s) which generates the isometric (and hence bounded) group given by T(t)f(s) = eist f(s). Then for every a> 0 and bE [0,1] we have 1

IIR(a + ib, A)lll = sE[O,l] sup I .b . I a+~ - ~s

1

a

1. Roundedness

85

which implies

a

1 00

-oo

IIR(a + ib, A)lll 2 db~ a

11

1 IIR(a + ib, A)lll 2 db= a o

-->

oo

as a--> 0 +.

So A does not satisfy condition (a) in Theorem 1.11 as well as condition (a) in Proposition 1.13. Since the same example works on the space L 00 [0, 1], we can take the pre• adjoint and see that also on the space 1 1 ([0, 1]) the operator A does not satisfy (a) in Theorem 1.11. It is still open to find analogous necessary and sufficient resolvent conditions for C0-semigroups on Banach spaces.

1.4 Polynomial boundedness In this subsection we introduce and discuss polynomial boundedness of Co-semi• groups. Surprisingly, this weaker notion allows a much simpler characterisation. Definition 1.15. A semigroup T(·) on a Banach space X is called polynomially bounded if IIT(t)ll ~ p(t), t ~ 0, for some polynomial p. In the following we will assume IIT(t)ll ~ K(1 + td) for some constants d ~ 0 (being not necessarily an integer) and K ~ 1. Note that every polynomially bounded semigroup T(·) still satisfies w0 (T) ~ 0. The following example shows that the converse implication does not hold. Example 1.16 (C0 -semigroups satisfying wo(T) ~ 0 with non-polynomial growth). The following construction is analogous to Example II.l.16 in the discrete case. Consider the Hilbert space

for some positive continuous function a satisfying a(O)

a(t + s)

~

a(t)a(s)

~

1 and

for all t, s E ~+

(III.3)

with the corresponding natural scalar product. On H take the right shift semigroup

T(·). We first check strong continuity ofT(·). For the characteristic function an interval [a, b] and t < b - a one has IIT(t)f- !II=

l

a

a+t

a 2 (s) ds

+

1b+t b

a2 (s)ds

---t

t-+0+

0.

f

on

86

Chapter III. Stability of C0 -semigroups

Further, for

f

EX we have by (III.3)

IIT(t)//1 2 = ioo lf(s- tWa 2(s) ds = :S a2 (t)

1 lf(s)l a (s) 00

2 2

1 00

lf(sWa 2(s + t) ds

a2 (t)IIJII 2

ds =

for every t 2': 0 and therefore IIT(t)ll :S a(t). A density argument shows that the semigroup T (·) is strongly continuous. Moreover, for the characteristic functions fn of the intervals [0, 1/n] we have

IIT(t)fnll 2 =

1.

t+.!. n

a2(s) ds =

t

and hence

1 rt+~ 2( 8 )d 8

IIT(t)ll 2 2': n Jt ~

1

a

fon a2(s) ds

2( a s) ds llfnll 2 0n a2(s) ds

rt+~ Jt 1

J

2( )

~ !!:..___!_,

n->oo a2(0)

So we obtain the norm estimate

:i~~ :S IIT(t)ll :S a(t) and hence the semigroup T(·) has the same growth as the function a. Note that if a(O) = 1, then IIT(t)ll = a(t) for all t 2': 0. Now every function a satisfying (III.3) and growing faster than every polynomial but slower than any exponential function with positive exponent yields an example of a non-polynomially growing Co-semigroup T(·) satisfying wo(T) :::; 0. As a concrete example of such a function take

a(t) := (t + 6)in(t+6) = ein2(t+6) (see Example !1.1.16), from which we obtain a C0 -semigroup growing like tint. Analogously, one can construct a C0 -semigroup growing as tin" t for any a 2': 1. The idea to use condition (III.3) belongs to Sen-Zhong Huang (private com• munication). The following theorem characterises operators generating polynomially boun• ded C0 -semigroups, see Eisner [64]. This generalises Theorem 1.11 for bounded C0-semigroups and a theorem of Malejki [183] (see also Kiselev [150]) in the case of Co-groups. The proof is a modification of the one of Theorem 1.11, see [64].

Theorem 1.17. Let X be a Banach space and A be a densely defined operator on X with s(A) :::; 0 and d E [0, oo). If the condition

J-oooo I

M

(R( a+ is, A) 2x, y} Ids :S --;:;:- (1

+a -d) llxiiiiYII

Vx E X, Vy E X' (III.4)

1. Roundedness

87

holds for all a > 0, then A is the generator of a Co-semigroup (T(t))t>O which does not grow faster than td, i.e., (III.5)

for some constant K and all t > 0. Conversely, if X is a Hilbert space, then the growth condition (III.5) implies (III.4) for the parameter d1 := 2d. Remark 1.18. Kiselev [150] showed that the exponent 2d in the implication (III.5) => (III.4) is sharp for the case of C0 -groups. Remark 1.19. Analogously to the bounded case, conditions for every x E X, for every y E X'

*

for some M :2: 1 and p, q > 1 satisfying ~ + = 1 are sufficient to obtain (III.4) and, for p = q = 2, equivalent to the generation of a polynomially bounded C 0 semigroup on Hilbert spaces.

If it is already known that A generates a Co-semigroup, then it becomes much easier to check whether the semigroup is polynomially bounded at least for a large class of semigroups. Following Eisner, Zwart [73], we say that an operator A has a p-integrable resolvent if for some/all a, b > s0 (A) the conditions

I: I:

IIR(a +is, A)xiiP ds < oo

'Vx EX,

(III.6)

IIR(b +is, A')YIIq ds < oo

'Vy EX'

(III. 7)

*

hold, where 1 < p, q < oo with ~ + = 1. Note that in particular condition (1.11) from Subsection 1.3.2 is satisfied for such semigroups by the Cauchy-Schwarz inequality. Plancherel's theorem applied to the functions t H e-atr(t)x and t H e-atT*(t)y for sufficiently large a > 0 implies that every generator of a Co• semigroup on a Hilbert space has 2-integrable resolvent. Moreover, for generators on a Banach space with Fourier type p > 1 condition (III.6) is satisfied auto• matically. Finally, every generator of an analytic semigroup (in particular, every bounded operator) on an arbitrary Banach space hasp-integrable resolvent for ev• ery p > 1. Intuitively, having p-integrable resolvent for some pis a good property of the generator A or/ and a good property of the space X.

Chapter III. Stability of Co-semigroups

88

Theorem 1.20 (Eisner, Zwart [73]). Let A be the generator of a C0 -semigroup (T(t))t>O having p-integrable resolvent for some p > 1. Assume that = {>. :

Cci

Re). > -0} is contained in the resolvent set of A and there exist a0 > 0 and M > 0 such that the following conditions hold.

M

(a) IIR(A,A)II:::; (Re>.)d

.

for some dE [O,oo) and all). wzth 0 < Re>. < a0 ;

(b) IIR(>.,A)II:::; M for all). with Re>. 2: ao. Then IIT(t)ll :::; K(l + t 2d-l) holds for some constant K > 0 and all t 2: 0. Conversely, if (T(t) )t>o is a Co-semigroup on a Banach space with IIT(t)ll :::; K(l + fY) for some constants ry 2: 0, K > 0 and all t 2: 0, then for every ao > 0 there exists a constant M > 0 such that the resolvent of the generator satisfies conditions (a) and (b) above for d = ry + 1. Proof. The second part of the theorem follows easily from the representation

The idea of the proof of the first part is based on the inverse Laplace transform representation of the semigroup presented in Subsection 1.3.2 and the technique from Zwart [264] and Eisner, Zwart [72]. We first note that by conditions (a) and (b) we obtain so (A) :::; 0. Next, since the function w f-+ R(a + iw, A)x is an element of LP(IR, X) for all x E X, we conclude by the uniform boundedness theorem that there exists a constant Mo > 0 such that IIR(a + i·, A)xiboR,X) :::; Mollxll

(III.8)

for all x EX. Similarly, one obtains the dual result, i.e., IIR(b + i·, A')yiiLq(JR,X') :S MollY II for ally EX'. Take now 0 < r

< a0 . By the resolvent equality we have

R(r + iw), A)x = [J +(a- r)R(r + iw, A)] R(a + iw, A)x. Hence IIR(r + iw, A)xll :::; [1 + Ia- riii(R(r + iw, A)IIJIIR(a + iw, A)xll :::;

[l+la-ri~]

IIR(a+iw,A)xil,

(III.9)

1. Roundedness

89

where we used (a). Combining this with the estimate (III.8), we find that

IIR(r + i·, A)xlboR,X) :S [1 + Ia- rl ~] Mollxll

:S M1 [ 1 + r1d] llxll·

(III.lO)

Similarly, we find that

(III.ll) From the estimates (III.lO) and (III.11) we obtain

/_: I(R(r + iw, A) 2 x, y) Idw = /_:

I(R(r + iw, A)x, R(r + iw, A)'y)l dw

:S IIR(r + i·, A)xiiLv(IR,X) IIR(r + i·, A')yiiLq(IR,X') :S M1MlllxiiiiYII [1 + r1d]

2

(III.12)

Convergence of the integral on the right-hand side of (III.12) implies that the inversion formula for the semigroup

T(t)x = - 1 27rt

1

00

. R(r e(r+M)t

+is, A) 2 x ds

-00

holds for all x E X by Theorem I.3.9. Notice that the condition r essential. Combining this formula with (III.12) we obtain

> s0 (A) is

1

erti(R(r + iw, A) 2 x, y)l dw I(T(t)x, y)l :S - 1 27rt _00 00

2 1 M1M111xiiiiYII [1 + d1 ] :S -ert

21rt

r

Since this holds for all 0 < r < a0 , we may chooser:= 1

-

(III.13)

t fort large enough giving d 2

I(T(t)x,y)l :S 27rteM1M1IIxiiiiYII [1+t] ·

(III.14)

So for large t the norm of the semigroup is bounded by Ct2d-l for some con• stant C. Since any C0 -semigroup is uniformly bounded on compact time intervals, D the result follows.

Chapter III. Stability of C 0 -semigroups

90

As mentioned above, every generator on a Hilbert space has 2-integrable resolvent, hence we have the following immediate corollary. Corollary 1.21. Let A generate a Co-semigroup (T(t))t>o on the Hilbert space H.

If A satisfies conditions (a) and (b) of Theorem 1.20 f(ir some d 2': 0 and a0 > 0, then there exists K > 0 such that IIT(t)ll ::; K[1 + t 2d-l] for all t 2': 0.

Remark 1.22. Notice that conditions (a) and (b) for 0 ::; d < 1 already imply

so(A) < 0 (use the power series expansion for the resolvent). On the other hand, for generators with p-integrable resolvent the equality w0 (T) = s0 (A) holds by Corollary 2.19. Combining these facts we obtain that in this case the semigroup is even uniformly exponentially stable. On the other hand, the exponential stability follows from the Theorem 1.20 only ford< ~·So for~ ::; d < 1 Theorem 1.20 does not give the best information about the growth of the semigroup. Nevertheless, for d = 1 the exponent 2d- 1 given in Theorem 1.20 is best possible (see Eisner, Zwart [72]). For d > 1 this is not clear. Note that the parameter d = 1 + 1 in the converse implication of Theorem 1.20 is optimal for 1 EN. Indeed, for X:= and

en

A ·.- (

010 ... 0) 0 0 1 ... 0

.

0 0 0 ... 0 conditions (a) and (b) in Theorem 1.20 are fulfilled ford= nand the semigroup generated by A grows exactly as tn-l. By Corollary 1.21 we see that the class of generators of polynomially bounded semigroups on a Hilbert space coincides with the class of generators of Co-semi• groups satisfying the resolvent conditions (a) and (b). For semigroups on Banach spaces this is not true since there exist C0 -semigroups with w0 (T) > s0 (A) (see Engel, Nagel [78, Examples IV.3.2 and IV.3.3]). As a corollary of Theorem 1.20 we have the following characterisation of polynomially bounded C0 -groups in terms of the resolvent of the generator. Theorem 1.23. Let A be the generator of a Co-group (T(t))tElR on a Banach space.

Assume that A hasp-integrable resolvent for some p > 1. Then the group (T(t))tElR is polynomially bounded if and only if the following conditions on the operator A are satisfied.

(a) cr(A) c i!R. (b) There exist a0 > 0 and d 2': 0 such that IIR(>.,A)II

constant M and all,\ with 0 < IRe .AI < ao.

(c) R(.X, A) is uniformly bounded on {A: IRe .AI 2: ao}.

M

< IRe.Aid for some

2. Uniform exponential stability

91

Proof. It is enough to show that the operator -A also hasp-integrable resolvent whenever A satisfies (a)-(c). Take any a > 0. Then by (b) or (c), respectively, R(A, A) is bounded on the vertical line -a+ iR By the resolvent equation we obtain

IIR(-a +is, A)xll

~

[1 + 2aiiR( -a+ is, A)IIJIIR(a +is, A)xll,

and therefore the function s f-t IIR( -a+ is, A)xll also belongs to LP(JR). The rest D follows immediately from Theorem 1.20. Again, this yields a characterisation of polynomially bounded C0-groups on Hilbert spaces. Note that the relation between the growth of the group and the growth of the resolvent appearing in (b) of Theorem 1.23 is as in Theorem 1.20.

2 Uniform exponential stability In this section we study the concept of uniform exponential stability of Co-semi• groups. Due to the unboundedness of the generator, this turns out to be more difficult than the corresponding notion for operators (compare Proposition II.1.3). Uniform exponential stability is defined as follows.

Definition 2.1. A Co-semigroup T(·) is called uniformly exponentially stable if there exist M ~ 1 and t=: > 0 such that

IIT(t)ll ~Me-et or, equivalently, if wo(T)

for all t ~ 0,

< 0.

For C0 -semigroups on finite-dimensional Banach spaces the classical Lya• punov theorem gives a simple characterisation in terms of the spectrum of the generator: A matrix semigroup (etA )t~o is uniformly exponentially stable if and only if ReA < 0 for every A E a( A), i.e., if s(A) < 0. However, on infinite• dimensional spaces this equivalence is no longer valid. The aim of this section is to characterise uniformly exponentially stable Co• semigroups on Banach and Hilbert spaces, preferably through spectral and resol• vent properties of the generator.

2.1

Spectral characterisation

The following elementary description of uniformly exponentially stable Co-semi• groups is the basis for many further results in this section, see Engel, Nagel [78, Prop. V.l.7].

Theorem 2.2. For a C0 -semigroup T(·) on a Banach space X the following asser• tions are equivalent.

(i) T(·) is uniformly exponentially stable, i.e., wo(T) < 0.

92

Chapter III. Stability of Co-semigroups

(ii) lim

t--+oo

(iii)

IIT(t)ll = 0.

IIT(to)ll < 1 for some to> 0.

(iv) r(T(to))

< 1 for some to> 0.

(v) r(T(t)) < 1 for all t > 0. The proof of the non-trivial implication (iv)=}(i) is based on the formula

r(T(t)) = etwo(T).

This theorem shows that, in particular, stability in the norm topology already implies uniform exponential stability. Furthermore, the equivalences (i){:}(iv){:}(v) show that in order to check uniform exponential stability it suffices to determine the spectral radius of T(to) for some to. Since, in most cases, the semigroup is unknown, this is not very helpful in general. However, if the spectrum of the semigroup, and therefore the spectral radius r(T(t)) can be obtained via some spectral mapping theorem from the spectrum of the generator, the analogue of the Lyapunov theorem holds. We devote the next section to this topic.

2.2

Spectral mapping theorems

The question is how to determine the spectrum and the growth bound of the semigroup using information about the generator only. For bounded generators we know that the semigroup is given as the expo• nential function etA (via the Taylor series or the Dunford functional calculus), and the spectral mapping theorem

a(T(t)) = eta(A)

for every t 2: 0

(III.15)

holds. This combined with r(T(t)) = etwo(T) implies the equality

s(A) = wo(T)

(III.16)

and allows one to check the uniform exponential stability by determining the spec• trum of the generator. It is well-known, see from e.g. Engel, Nagel [78, Examples IV.2.7, IV.3.3-4], neither (III.15) nor (III.16) holds in general. We give a short survey of why and how the spectral mapping theorem (III.15) fails, and under which assumptions and modifications it becomes true. As a first modification, we should exclude the point 0 in (III.15) since it can occur in the left-hand, but not in the right-hand term. So we choose the following terminology.

Definition 2.3. A C0 -semigroup T(·) on a Banach space X with generator A sat• isfies the spectral mapping theorem (or has the spectral mapping property) if a(T(t)) \ {0} = eta(A)

for every t 2: 0.

(III.17)

93

2. Uniform exponential stability

For semigroups satisfying the spectral mapping theorem, equality (III.16) holds automatically by the formula r(T(t)) = etwo(T). Note that the spectral mapping theorem always holds for the point and resid• ual spectrum, see Engel, Nagel [78, Theorem IV.3.7]. Moreover, the spectral map• ping theorem holds for certain quite large and important classes of C0 -semigroups.

Theorem 2.4. (see Engel, Nagel [78, Cor. IV.3.12]) The spectral mapping theorem holds for all eventually norm continuous Co-semigroups, hence for the following classes of semigroups:

(i) uniformly continuous semigroups (or, equivalently, for semigroups with bounded generators), (ii) eventually compact semigroups, (iii) analytic semigroups, (iv) eventually differentiable semigroups.

In particular, for any such semigroup equality (III.16) holds, implying that the semigroup is uniformly exponentially stable if and only if s(A) < 0. For the proof of the first assertion we refer to Engel, Nagel [78, Cor. IV.3.12]. Equality (III.16) is then a consequence of the formula r(T(t)) = etwo(T)_ However, the following example shows that even for simple multiplication semigroups the spectral mapping theorem can fail. Example 2.5. Let T( ·) be the multiplication semigroup on l 2 given by

for qn :=~+in. Then

1 E a(T(2n)) = {e21r/n: n EN}, while

1~

e27rcr(A) = { e27r /n

: n E N}.

So the spectral mapping theorem (III.17) does not hold. We now look for some weaker forms of a spectral mapping theorem still implying (III.16). The first is motivated by the above example.

Definition 2.6. A C0 -semigroup T(·) with generator A satisfies the weak spectral mapping theorem (or has the weak spectral mapping property) if a(T(t)) \ {0} = etcr(A) \ {0} for every t 2': 0.

(III.18)

Chapter III. Stability of C0 -semigroups

94

Every multiplication semigroup on spaces C 0 (0) and £P(O, f.l) satisfies the weak spectral mapping theorem, see Engel, Nagel [78, Prop. IV.3.13]. Conse• quently, we obtain (by the spectral theorem) that every semigroup of normal op• erators on a Hilbert space has the weak spectral mapping property. Moreover, the weak spectral mapping theorem holds for bounded C 0 -groups on Banach spaces, see Engel, Nagel [78, Prop. IV.3.13], and, more generally, for every C 0 -group with so-called non-quasianalytic growth, see Huang [131] and Huang, Nagel [197] for details. We finally present another weaker spectral mapping property, still sufficient for equation (III.16). Definition 2.7. A Co-semigroup T(·) with generator A satisfies the weak circular spectral mapping theorem (or has the weak circular spectral mapping property) if

f · a(T(t)) \ {0}

=f

· et 0 satisfying IIT(t)ll :=:; Mewt for every t 2 0. Then 1t 1- e-pwt e-pwsiiT(s)T(t- s)xiiP ds ---IIT(t)xiiP =

pw

0

:=:; MP !at IIT(t- s)xiiP ds :=:; MP

fooo IIT(s)xiiP ds

holds for every x EX and t 2 0, where the right-hand side is finite by assumption. Hence, every orbit {T(t)x: t 2 0} is bounded and therefore the semigroup T(·) is bounded by the uniform boundedness principle, hence L := supt>o IIT(t)ll < oo. We further observe that

tiiT(t)xiiP =!at IIT(t- s)T(s)xiiP ds :=:; LP

fooo IIT(s)xiiP ds 1

for every x EX and t 2 0. This implies boundedness of the set {t"iiT(t)x, t > 0} 1 for every x EX. By the uniform boundedness principle, supt>o t"P IIT(t)ll < oo and 0 therefore T(·) is uniformly exponentially stable by Theorem-2.2 (iii). Datko [55] proved this theorem for p = 2 and Pazy [209, Theorem 4.4.1] extended it to the case p 2 1. There are various generalisations of the Datko-Pazy theorem. As an exam• ple we state theorems due to van Neerven and Rolewicz, see van Neerven [204, Corollary 3.1.6 and Theorem 3.2.2].

Theorem 2.11. LetT(·) be a Co-semigroup on a Banach space X, p E [1,oo) and {3 E Lfoc(JR+) a positive function satisfying

1 00

If

1

00

{3(t)dt =

,B(t)IIT(t)xiiP dt < oo

00.

for all x EX,

Chapter III. Stability of Co-semigroups

96

then T (·) is uniformly exponentially stable. Theorem 2.12 (Rolewicz). LetT(·) be a Co-semigroup on a Banach space X. If there exists a strictly positive increasing function ¢ on IR+ such that

1 00

¢(JJT(t)xJI) dt < oo for all x E X, JJxJJ :::; 1,

then T (·) is uniformly exponentially stable. For further discussion of the above result we refer to van Neerven [204, pp. 110-111]. The following theorem is a weak version of the Datko-Pazy theorem due to Weiss [255]. Theorem 2.13 (Weiss). LetT(·) be a Co-semigroup on a Hilbert space X. If for

somepE [1,oo),

1 00

J(T(t)x,y)JPdt < oo for all x EX and y EX',

then T (·) is uniformly exponentially stable. We refer to Weiss [256] for a discrete version of the above result. Remark 2.14. The assertion of Theorem 2.13 also holds for bounded semigroups on general Banach spaces, see van Neerven [204, Theorem 4.6.3]. However, without the boundedness assumption it fails and wo(T) > 0 becomes possible even for positive semigroups, see van Neerven [204, Theorem 4.6.5 and Example 1.4.4]. We refer to Tomilov [242] and van Neerven [204, Section 4.6] for other generalisations of Theorem 2.13.

For further generalisations of the Datko-Pazy theorem see, e.g., Vii [252], van Neerven [204, Sections 3.3-4] and Theorem 3.9 below.

2.4 Theorem of Gearhart For C0-semigroups on Hilbert spaces there is a useful characterisation of uniform exponential stability in terms of the generator's resolvent on the right half plane. This is the classical theorem of Gearhart [94] and can be considered as a gener• alisation of the finite-dimensional Lyapunov theorem to the infinite-dimensional case. To this theorem and its generalisations this subsection is dedicated. We begin with the following result on uniform exponential stability using the inverse Laplace transform method presented in Subsection 1.3.2. Theorem 2.15. Let A generate a Co-semigroup T(-) on a Banach space X such

that the resolvent of A is bounded on the right half plane {z : Re z > 0}. Assume further that /_: J(R 2 (is,A)x,y)Jds < oo

for all x EX andy EX'.

(111.21)

2. Uniform exponential stability

97

Then T (·) is uniformly exponentially stable. Proof. We first observe that, since the resolvent is bounded on the right half plane, it is also bounded on a half plane { z : Re z > -8} for some o> 0 by the power series expansion of the resolvent and hence so(A) < 0. Therefore, by condition (111.21) and Theorem 1.3.8 we have (T(t)x, y) = -21 7rt

100

.

-oo

e~st(R 2 (is, A)x, y)

ds

for all x EX, y E X'.

The uniform boundedness principle now implies

1 I(T(t)x,y)l :S -21 100 I(R 2 (is,A)x,y)lds :S -MIIxiiiiYII 27rt 7rt -00 for some constant M and all x E X, y E X', hence II T (t) II :S 2~t t-+

------t

00.

0 as 0

Note that this result generalises a result of Xu and Feng [259]. We now present Gearhart's theorem for which we give two proofs: one based on the theorem above, i.e., on the properties of the inverse Laplace transform, and the second using the Datko-Pazy theorem.

Theorem 2.16 (Gearhart, 1978). Let A generate a Co-semigroup T(·) on a Hilbert space H. Then T(·) is uniformly exponentially stable if and only if there exists a constant M > 0 such that

IIR(A,A)II < M for all,\ with Re,\ > 0.

(III.22)

Proof I. As in the proof of Theorem 2.15 we see, by the power series expansion for the resolvent, that s0 (A) < 0 holds. By Theorem 2.15 it suffices to check condition (III.21). Take a > w 0 (T) and x, y E H. By the Laplace transform representation of the resolvent of A and Plancherel's theorem applied to the function t f-+ e-atT(t)x we obtain

/_: IIR(a +is, A)xll 2 ds =

100

e- 2atiiT(t)xll 2 dt < oo.

Further, by (111.22) and the resolvent identity, the estimate

IIR(is, A)xll

=

II[I + aR(is, A)]R(a +is, A)xll :S [1 + aMJIIR(a +is, A)xll

I:

holds for every s E lR and hence

IIR(is, A)xll 2 ds < oo.

98

Chapter III. Stability of C0 -semigroups

Applying the same arguments to the operator A* and the function t obtain

f----7

T*(t)y we

/_: I(R 2 (is,A)x,y)ids = /_: I(R(is,A)x,R(-is,A*)y)ids

~

I

(/_: IIR(is,

A)xll 2

ds)

2

I

(/_:

IIR(is,

2 A*)yll 2 ds)

< oo,

and the theorem is proved.

0

Proof II. (Weiss [255]) Assume wo := wo(T) 2 0 and take x E H and a, ao with ~ wo oo IIT(t)xll = 0 for every x EX. The first example below is, in a certain sense, typical on Hilbert spaces, see Theorem 3.11 below.

Example 3.2. (a) (Shift semigroup). Consider H := L2 (1R+, H 0 ) for a Hilbert space Ho and T(·) defined by

T(t)f(s)

:=

f(s

+ t),

f E H, t, s ~ 0.

(III.23)

The semigroup T(·) is called the left shift semigroup on Hand is strongly stable. Note that the spectrum of its generator is the whole left halfplane. The same semigroup on the spaces Co(lR+, X) and LP(JR+, X), X a Banach space, is also strongly stable for 1 :S p < oo, but not for p = oo. (b) (Multiplication semigroup, see Engel, Nagel [78, p. 323]). Consider X:= C 0 (0) for a locally compact space n and the operator A given by

Af(s) := q(s)f(s),

f EX, s E 0,

with the maximal domain D(A) := {! E X : qf E X}, where q is a continuous function on 0. The operator A generates the C 0 -semigroup given by

T(t)f(s) =

etq(s)

f(s),

f EX, s E 0,

if and only if Re q is bounded from above. The semigroup is bounded if Re q( s) :S 0 for every s E n. Moreover, the semigroup is strongly stable if and only if Re q( s) < 0 for every s E 0. Indeed, if Re q( s) < 0, then

I!T(t)fl! :S sup etReq(s)llfll sEK

for every function

f

------7

t--+oo

0

with compact support K. By the density of these functions

T(·) is strongly stable. Conversely, if q(so) E ilR, then IIT(t)fll ~ lf(so)l for every f EX and hence T(·) is not strongly stable. Note that a( A) = q(O) and therefore every closed set contained in the closed left halfplane can occur as the spectrum of the generator of a strongly stable Co-semigroup. For a concrete example of strongly stable and strongly convergent semigroups appearing in mathematical models of biological processes see, e.g., Bobrowski [36], Bobrowski, Kimmel [37, 38]. The following property of strongly stable semigroups follows directly from the uniform boundedness principle.

101

3. Strong stability

Remark 3.3. Every strongly stable Co-semigroup T(·) is bounded, hence O"(A) C {z: Rez :S 0} holds for the generator A. Note that conditions Po-(A) n ilR = 0 and Po-(A') n ilR = 0 are also necessary for strong stability by the spectral map• ping theorem for the point and residual spectrum, see Engel, Nagel [78, Theorem

IV.3.7]. We now state an elementary but very useful property implying strong stability. Lemma 3.4. LetT(·) be a bounded C0 -semigroup on a Banach space X and let

xEX.

(a) If there exists an unbounded sequence {tn} ~= 1 C IR+ such that limn---;oo IIT(tn)xll = 0, then limt---;oo IIT(t)xll = 0. (b) IJT(·) is a contraction semigroup, then lim IIT(t)xll exists. f---;oo

Proof. The second part follows from the fact that for contraction semigroups the function t f---7 IIT(t)xll is non-increasing. To verify (a) assume that IIT(tn)xll ---+ 0 and take E > 0, n EN such that IIT(tn)xll < c: and M := SUPt:;:o:o IIT(t)ll· We obtain IIT(t)xll :S IIT(t- tn)IIIIT(tn)xll oo t supt~O t

0

0

1 ds = 0, ds < oo for ally EX'.

In particular, T( ·) is strongly stable if and only if (b) holds for all x E X. The proof is an easy modification of the one of Theorem 1.6. Remark 3.8. Note that for bounded Co-semigroups the second part of condition (b) holds automatically.

We finally state a continuous analogue of the result of Muller (Theorem 11.2.8) on the asymptotic behaviour of semigroups which are not uniformly exponentially stable, see van Neerven [204, Lemma 3.1.7]. Theorem 3.9 (van Neerven). LetT(·) be a Co-semigroup on a Banach space X with w 0 (T) ~ 0. Then for every function a: JR.+~ [0, 1) converging monotonically to 0 there exists x E X with llxll = 1 such that

IIT(t)xll ~ a(t)

for every t

~ 0.

Theorem 3.9 means that strongly stable semigroups which are not exponen• tially stable possess arbitrarily slowly decreasing orbits. Strong stability of semigroups on Hilbert spaces can be characterised in terms of strong stability of its cogenerator, see Subsection V.2.5. A necessary and suf• ficient condition for strong stability on Hilbert spaces using the resolvent of the generator is given in Subsection 3.4 below.

3.2

Representation as shift semigroups

We now give a representation of strongly stable semigroups following directly from Theorem 1.9.

(a) Every strongly stable contraction semigroup on a Banach space X is isometrically isomorphic to the left shift semigroup on a closed subspace of Co (JR.+, X).

Proposition 3.10.

3. Strong stability

103

(b) Every strongly stable Co-semigroup on a Banach space X is isomorphic to the left shift semigroup on a closed subspace of Co(lR.+, X1 ), where X1 =X endowed with an equivalent norm. In analogy to Theorem II.2.11, Lax and Phillips showed that Example 3.2 (a) represents the general situation for contractive strongly stable C0 -semigroups on Hilbert spaces. Theorem 3.11 (Lax, Phillips [164, p. 67], see also Lax [163, pp. 45o-451]). Let

T(·) be a strongly stable contraction semigroup on a Hilbert space H. Then T(·) is unitarily isomorphic to a left shift, i.e., there is a Hilbert space H0 and a unitary operator U : H --t H1 for some closed subspace H1 C L 2 (JR.+, Ho) such that UT(-)U- 1 is the left shift on H1. Proof. The idea is again to identify a vector x with its orbits Strong stability ofT(·) implies the equality

t"J

['YO d

llxll 2 =- Jo

H

T(s)x.

ds(IIT(s)xll 2 )ds= Jo (-2Re(AT(s)x,T(s)x))ds

(III.24)

for every x E D(A). Define now the new seminorm llxll~ := -2Re (Ax, x)

on D(A),

which is non-negative by the Lumer-Phillips theorem. Note that it comes from the scalar semiproduct (x, y)y := -(Ax, y) - (x, Ay). Consider the subspace H0 := {x E D(A): llxlly = 0} with its completion Y := (D(A)/Ho, II ·IIYf. We now define the operator J : H --t L 2 (Y) by

(Jx)(s) := T(s)x,

x E H, s 2 0,

which is an isometry by equality (III.24). Therefore it is unitary from H to its (closed) range. Since the semigroup (JT(t)J- 1 )r~ 0 is the right shift semigroup on 0 rg J, the proof is complete. Note that in contrast to the analogous Proposition 3.10, the above representation respects the Hilbert space structure.

3.3 Spectral conditions In this subsection we discuss spectral conditions on the generator implying strong stability of the semigroup. Analogously to the discrete case, these conditions use "smallness" of the spectrum of the generator on the imaginary axis. The first result is the stability theorem for C0 -semigroups proved by Arendt, Batty [9] and Lyubich, Vii [180] independently, generalising a result of Sklyar, Shirman [234] on semigroups with bounded generators. The discrete version of this result has been treated in Subsection II.2.3.

104

Chapter III. Stability of Co-semigroups

Theorem 3.12 (Arendt-Batty-Lyubich-Vii, 1988). Let T(·) be a bounded semi•

group on a Banach space X with generator A. Assume that

(i) Pa(A') n ilR = 0; (ii) (j(A) n ilR is countable. Then T(·) is strongly stable. The proof based on the Lyubich and Vii construction of the isometric limit semigroup, which is a natural modification of the discrete case, can be found in Engel, Nagel [78, pp. 263-264 and Theorem V.2.21]. For the alternative proof by Arendt and Batty using the Laplace transform see Arendt, Batty [9] or Arendt, Batty, Hieber, Neubrander [10, Theorem 5.5.5]. For generalisations and extensions of Theorem 3.12 see Batty, Vu [27], Batty [22], Batty, van Neerven, Rabiger [26]. For a concrete application of the above result to semigroups arising in biology see Bobrowski, Kimmel [37]. Remark 3.13. 1) Conditions (i) and (ii) in the above theorem are of different nature. The first one is necessary for strong stability while the second is a useful, but restrictive assumption. 2) For weakly relatively compact semigroups, in particular for bounded semi• groups on reflexive Banach spaces, condition (i) is equivalent to

by the mean ergodic theorem, see Subsection !.1.6. An immediate corollary of the above theorem is the following simple spectral criterion (see also Chill, Tomilov [50] for an alternative proof using a theorem of Ingham on the Laplace transform).

Corollary 3.14. Let A generate a bounded Co-semigroup T(·) on a Banach space. If (j(A) n ilR = 0, then T(·) is strongly stable.

Proof. The assertion follows from Pa(A') n ilR = Ra(A) n ilR = Batty-Lyubich-Vu theorem.

0 and the Arendt• 0

Theorem 3.12 is particularly useful if one combines it with the Perron• Frobenius spectral theory for positive semigroups on Banach lattices (see Nagel (ed.) [196] and Schaefer [227]).

Corollary 3.15. Let A generate a bounded, eventually norm continuous, positive semigroup T(·) on a Banach lattice X. Then T(·) is strongly stable if and only if 0 ~ Pa(A').

Proof. By the Perron-Frobenius theory, the spectrum of the generator of a posi• tive bounded semigroup on the imaginary axis is additively cyclic, i.e., ia E o-(A) implies iaZ E (j(A) for real a, see Nagel (ed.) [196, Theorem C-III.2.10 and Propo• sition C-III.2.9]. Moreover, the spectrum of the generator of an eventually norm

3. Strong stability

105

continuous semigroup is bounded on every vertical line, see Engel, Nagel [78, The• orem 11.4.18]. The combination of these two facts leads to a(A) n i!R c {0}. The D theorem of Arendt-Batty-Lyubich-Vu concludes the argument. Remark 3.16. We note that in the above result the condition 0 ~ P,.(A') cannot

be replaced by 0 ~ P,.(A). This can be easily seen for A:= Tr-Ion l 1 , where Tr is the right shift operator on l 1 .

Moreover, the following is a further direct corollary of Theorem 3.12 and the mean ergodic theorem. Corollary 3.17. Let T(·) be a bounded holomorphic Co-semigroup on a Banach

space with generator A. Then T(·) is strongly stable if and only ifO

~

P,.(A').

For an alternative proof see Bobrowski [35]. As in the discrete case, the theorem of Arendt-Batty-Lyubich-Vu can be extended to completely non-unitary contraction semigroups on Hilbert spaces. Theorem 3.18. (see Foia§ and Sz.-Nagy [238, 11.6. 7] and Kerchy, van Neerven [148])

LetT(·) be a completely non-unitary contractive C0 -semigroup on a Hilbert space with generator A. If a( A) n i!R has Lebesgue measure 0,

then T(-) and T*(-) are both strongly stable. See also Kerchy, van Neerven [148] for related results. The following example shows that in the above theorem one cannot replace a completely non-unitary by a contractive semigroup with P,.(A) n i!R = 0. The idea of this elegant construction belongs to David Kunszenti-Kovacs (oral com• munication). Example 3.19. Take the Cantor set C on [0, 1] constructed from the intervals

(a1,b1) = (~, ~), (a2,b2) = (!, ~), (a3,b3) = (~,~)and so on. Take further the Cantor set 6 constructed from the intervals { (a-n, b:)} such that 6 has measure ~· By the natural linear transformation mapping each (an, bn) onto (an, bn) and its continuation we obtain a bijective monotone map j : 6 ----+ C being an analogue of the Cantor step function. The image of the Lebesgue measure J.L under j we denote by J.li· We observe that by construction J.LJ is continuous. Note that, since J.lj (C) = ~, it is not absolutely continuous. Consider H := L 2 (C,J.Lj) and the bounded operator A on H defined by Af(s) := isf(s). Note that the corresponding unitary group is defined by T(t)f(s) = eist f(s). Since J.l,j is continuous, P,.(A') = P,.(A) = 0. Moreover, a(A) = iC and therefore J.L(a(A)) = 0. However, T(·) is unitary and hence limt-too IIT(t)xll = 0 or limt-->oo IIT*(t)xll = 0 implies x = 0.

We finish this subsection by the observation that a(A) ni!R for the generator A of a strongly stable semigroup can be an arbitrary closed subset of i!R, see Example 3.2(b). So, "smallness" conditions on the boundary spectrum are far from being necessary for strong stability.

Chapter III. Stability of C0 -semigroups

106

3.4 Characterisation via resolvent We now present a powerful resolvent approach to strong stability introduced by Tomilov [243]. In contrast to the spectral approach from the previous subsection, resolvent conditions are necessary and sufficient at least for C0 -semigroups on Hilbert spaces.

Theorem 3.20 (Tomilov). Let A generate a C0 -semigroup T(·) on a Banach space X satisfying so (A) ~ 0 and x E X. Consider the following assertions. (a)

lim a

a---+0+

1 00

_ 00

IIR(a +is, A)xll 2 ds = 0,

00

limsupa1 IIR(a +is, A')yll 2 ds a---+0+ -oo

< oo for ally EX';

(b) lim IIT(t)xll = 0. t---+oo

Then (a) implies (b). Moreover, if X is a Hilbert space, then (a){:}(b). In particular, condition (a) for all x EX implies strong stability ofT(·) and, on Hilbert spaces, is equivalent to it. Proof. We first prove that (a)=?(b). By Theorem !.3.9 (see also Theorem !.3.8) and the Cauchy-Schwarz inequality we have

1

eat oo I(T(t)x,y)l ~I(R 2 (a+is,A)x,y)lds 2nt _ 00

~

;:: (j_:

1

IIR(a +is,

A)xll 2

ds)

2

(j_:

1

IIR(a +is,

A')yll 2

ds)

2

for every t > 0, a> 0 andy EX'. By (a) and the uniform boundedness principle there exists a constant M > 0 such that

a

i:

IIR(a +is, A')YII 2 ds

~ M 2 IIYII 2

Therefore, we have IIT(t)xll ~ -Meat ( a

2nta

1

00

_ 00

for every y EX' and a> 0.

IIR(a +is, A)xll 2 ds )

!

(III.25)

t'

By choosing a:= we obtain by (III.25) limt---+oo IIT(t)xll = 0. Assume now that T(·) is strongly stable on a Hilbert space X. By Parseval's equality

a

1:

IIR(a +is, A)xll 2 ds =a

1

00

e- 2atiiT(t)xll 2 dt,

where the right-hand side is one half times the Abel mean of the function t r---+ IIT(t)xll 2 . Therefore it converges to zero as a-----* 0+ by the strong stability ofT(·). This proves the first part of (a). The second condition in (a) follows from Theorem 1.11. D

3. Strong stability

107

Remark 3.21. Corollary 2.19 shows that on Banach spaces the integrals appearing in (a) above do not converge in general. We now show that in the above theorem (b) does not imply (a) in general even for bounded generators for which all the integrals in (a) converge. This is the following example which is a modification of Example 1.14. Example 3.22. Take a bounded sequence {an}~=l C { z : Re z < 0} such that {an : n E N} nilR = i [-1, 1]. Consider X = l 1 and the bounded operator A given by

A(x 1 , x 2, ... ) = (a 1 x 1 , a2x 2, ... ) on X. The semigroup generated by A is strongly

stable by Example 3.2 (b). The adjoint of A is given by A'(x 1 , x 2, ... ) = (a 1 x1, a 2x 2 , ... ) on zoo. So we have for every a> 0, JbJ < 1 andy E U!(l), 1 JJR(a + ib, A')yJJ =sup JxnJ > nEN Ja+ib-anJ- 2dist(a+ib,i[-1,1]) Therefore we obtain

a

1

00

-oo

JJR(a + ib, A')yJJ 2 db~ a

1 2a

1 1

- 12 db= -1 ---too as a---t 0+, _ 1 4a 2a

so the second part of (a) in Theorem 3.20 does not hold for y from the open set U1 (1) in X'. 2

By Theorems 1.11 and 3.20 one immediately obtains the following charac• terisation of strong stability in the case of bounded C0 -semigroups on Hilbert spaces.

Corollary 3.23. Let A generate a bounded semigroup T (·) on a Hilbert space H and x EX. Then JJT(t)xJJ---t 0 if and only if lim a

a-->0+

1

00

_

JJR(a +is, T)xll 2 ds = 0.

(III.26)

00

In particular, T(·) is strongly stable if and only if (III.26) holds for every x in a dense set of H. It is still an open question whether the above characterisation holds for Co• semigroups on Banach spaces.

Remark 3.24. There are more results on strong stability of Ca-semigroups and their connection to the behaviour of the generator's resolvent. As some recent contributions we mention papers by Batkai, Engel, Pruss, Schnaubelt [20], Batty, Duyckaerts [25] and Borichev, Tomilov [39] on polynomial decay of the orbits and Chill, Tomilov [49] on characterisations of strong stability on Banach spaces with Fourier type. For so-called pointwise resolvent conditions see Tomilov [243] and Batty, Chill, Tomilov [24]. We recommend the excellent overview by Chill, Tomilov [50] for further results and references.

Chapter III. Stability of Co-semigroups

108

4

Weak stability

In this section we consider stability of C0 -semigroups with respect to the weak operator topology. As in the discrete case, this turns out to be much more difficult than its strong and uniform analogues.

4.1

Preliminaries

We begin with the definition and some examples.

Definition 4.1. A Co-semigroup T(·) on a Banach space X is called weakly stable iflimt--. 00 (T(t)x,y) = 0 for every x EX andy EX'. Note that by the uniform boundedness principle every weakly stable semi• group T(·) on a Banach space is bounded, hence wo(T) :::; 0 holds. In particular, the spectrum of the generator A belongs to the closed left half plane. Moreover, the spectral conditions Pu(A) n i!R = 0 and Ru(A) n i!R = Pu(A') n i!R = 0 are necessary for weak stability. (a) The left and right shift semigroups are weakly stable and iso• metric (and hence not strongly stable) on the spaces C0 (1R, X) and LP(IR, X) for every Banach space X and 1 < p < oo.

Example 4.2.

(b) The right shift semigroup on £P (IR+, X) for a Banach space X defined by

(T(t)f)(s) = {f(s- t), 0,

s 2 t, s 0 such that M intersects every sub-interval of JR+ of length f. It turns out that weak convergence to zero for such a sequence already implies weak stability.

Theorem 4.4. LetT(·) be a Co-semigroup and suppose that weak-limn-+oo T(tn) = 0 for some relatively dense sequence {tn} ~ 1 . Then T( ·) is weakly stable.

Proof. Without loss of generality, by passing to a subsequence if necessary, we assume that {tn};::'=l is monotone increasing and set f := supnEN(tn+l - tn), which is finite by assumption. Since every Co-semigroup is bounded on compact time intervals and (T(tn))nEN is weakly converging, hence bounded, we obtain that the semigroup (T(t))t?:O is bounded. Fix x EX, y EX'. FortE [tn, tn+l] we have

(T(t)x, y) = (T(t- tn)x, T' (tn)Y), where (T'(t))t>o is the adjoint semigroup. We note that by assumption T'(tn)Y----+ 0 in the weak*-topology. Further, the set Kx := {T(s)x: 0:::; s:::; £}is compact in X and T(t-tn)x E Kx for every n E N. Since pointwise convergence is equivalent to the uniform convergence on compact sets (see, e.g., Engel, Nagel [78, Prop. A.3]), we see that D limt-+oo(T(t)x, y) = 0.

Remark 4.5. Taking tn = n in the above theorem, we see that a Co-semigroup T(·) is weakly stable if and only if the operator T(1) is weakly stable. This builds a bridge between weak stability of discrete and continuous semigroups, and we will return to this aspect later in Section V.l.

Remark 4.6. One cannot drop the relative density assumption in Theorem 4.4 or even replace it by the assumption of density 1, see Section 5 below. We finally mention that by Theorem 11.3.8, weak orbits of a Co-semigroup

T( ·) which is not exponentially stable can decrease arbitrary slowly. More precisely, for every positive function a : JR+ ----+ JR+ decreasing to zero, there exist x E X andy EX' such that the corresponding weak orbit satisfies I(T(tj)x,y)l 2 a(tj) for some sequence {tj}~ 1 converging to infinity and all j EN. However, it is not

clear whether the result of Badea, Miiller [12] holds in the continuous case, i.e., whether one may replace {tj}~ 1 by JR+ for weakly stable semigroups.

4.2

Contraction semigroups on Hilbert spaces

In this subsection we present some classical theorems on the decomposition of contractive C0 -semigroups on Hilbert spaces with respect to their qualitative be• haviour.

Chapter III. Stability of C0 -semigroups

110

We begin with the decomposition into unitary and completely non-unitary parts due to Foia§ and Sz.-Nagy [237].

Theorem 4.7 (Foia§, Sz.-Nagy). LetT(·) be a contmction semigroup on a Hilbert space H. Then His the orthogonal sum of two T(·)- andT*(·)-invariant subspaces H1 and H2 such that (a) H1 is the maximal subspace on which the restriction Tt(·) ofT(-) is unitary;

(b) the restrictions of T (·) and T* (-) to H2 are weakly stable. We present the proof of Foguel given in [86] being analogous to the proof of the discrete version of this result (Theorem II.3.9).

Proof. Define

H1 := {x E H: IIT(t)xll = IIT*(t)xll = llxll for all t ~ 0}. Observe that for every 0 i= x E H1 and t ~ 0, llxll 2 = (T(t)x, T(t)x) = (T*(t)T(t)x, x) :S IIT*(t)T(t)xllllxll :S llxll 2 . Therefore, by the equality in the Cauchy-Schwarz inequality and the positivity of llxll 2 , we obtain T*(t)T(t)x = x. Analogously, T(t)T*(t)x = x. On the other hand, every x with these two properties belongs to H1. So we proved the equality

H1 = {x E H: T*(t)T(t)x = T(t)T*(t)x = x for all t

~

0}

(III.27)

which shows, in particular, that H 1 is the maximal (closed) subspace on which

T(·) is unitary. The T(t)- and T*(t)-invariance of H1 follows from the definition of H1 and the equality T*(t)T(t) = T(t)T*(t) on H 1 . To show (b) take x E H2 := Hf. We first note that H 2 is T(·)- and T*(·)• invariant since H1 is so. Suppose now that T(t)x does not converge weakly to

oo, or, equivalently, that there exists y E H, c > 0 and a sequence {tn}~=l such that I(T(tn)x,y)l ~ c for every n EN. On the other hand, there exists a weakly converging subsequence of {T(tn)x }~=l· For convenience we denote the subsequence again by {tn}~=l and its limit by xo. The closedness and T(t)• invariance of H 2 imply that x 0 E H2. For a fixed t 0 ~ 0 we obtain zero as t

----+

IIT*(to)T(to)T(t)x- T(t)xll 2 =

IIT*(to)T(t + to)xll 2

-

2(T*(to)T(t + to)x, T(t)x) + IIT(t)xll 2

:S IIT(t + to)xll 2 - 2IIT(t + to)xll 2 + IIT(t)xll 2 = IIT(t)xll 2 -IIT(t

+ to)xll 2 ·

The right-hand side converges to zero as t ----+ oo since the function t H IIT(t)xll is monotone decreasing on IR+. Therefore we obtain limt_, 00 IIT*(to)T(to)T(t)x• T(t)xll = 0.

111

4. Weak stability

We now remember that weak-limn-HXl T(tn)x = xo. This implies immedi• ately that weak-limn-+oo T*(to)T(to)T(tn)x = T*(to)T(to)xo. By the considera• tions above, we have on the other hand that weak-limn-+oo T*(to)T(to)T(tn)x = Xo and therefore T*(to)T(to)xo = xo. One shows analogously that T(to)T* (t 0 )x0 = x 0 and xo E H1. By H1 U H2 = {0} this implies xo = 0, which is a contradiction. Analogously one shows that the restriction ofT*(-) to H2 converges weakly to zero as well. 0 Remark 4.8. The restriction ofT(·) to the subspace H2 in Theorem 4.7 is com•

pletely non-unitary (c.n. u. for short), i.e., there is no subspace of H2 on which the restriction ofT(-) becomes unitary. In other words, Theorem 4. 7 states that every Hilbert space contraction semigroup can be decomposed into a unitary and a c.n.u. part and the c.n.u. part is weakly stable. For a systematic study of completely non-unitary semigroups as well as an alternative proof of Theorem 4. 7 (except weak stability on H2) using unitary dilation theory see the monograph of Sz.-Nagy and Foia§ [238, Proposition 9.8.3].

On the other hand, the following theorem gives a decomposition into weakly stable and weakly unstable parts due to Foguel [86]. We give a simplified proof of it. Theorem 4.9 (Foguel). LetT(·) be a contraction semigroup on a Hilbert space H.

Define W := {x E H: lim(T(t)x, x) = 0}. t--->0

Then W = {x E H: lim T(t)x = 0 weakly}= {x E H: lim T*(t)x = 0 weakly}, t--->0

t--->0

W is a closed T(·)- and T*(·)-invariant subspace of H and the restriction ofT(·) to W j_ is unitary.

Proof. We first show that weak-limt-+oo T(t)x = 0 for a fixed x E W. By Theorem 4.7 we may assume that x E H 1 . If we takeS:= lin{T(t)x: t 2: 0}, then by the decomposition H = S EB Sj_ it is enough to show that limt-+oo (T(t)x, y) = 0 for all y E S. For y := T(t 0 )x we obtain (T(t)x, y) = (T*(t 0 )T(t)x, x) = (T(t- t 0 )x, y)

-->

0 for to :S t--> oo,

since the restriction ofT(·) to H 1 is unitary. The density of lin{T(t)x : t 2: 0} inS implies limt-+oo (T(t)x, y) = 0 for every y E S and therefore weak-limt-+oo T(t)x = 0. Analogously, weak-limt-+oo T*(t)x = 0. The converse implication, the closedness and the invariance of W are clear. The last assertion of the theorem follows directly from Theorem 4. 7. 0 Combining Theorem 4. 7 and Theorem 4.9 we obtain the following decompo• sition into three orthogonal subspaces.

Chapter III. Stability of C0 -semigroups

112

Theorem 4.10. LetT(·) be a contraction semigroup on a Hilbert space H. Then H

is the orthogonal sum of three closed T(·)- and T*(·)-invariant subspaces H 1 , H 2 and H3 such that the restrictions T1(·), T2(·) and T3(·) satisfy the following.

(a) Tt(·) is unitary and has no non-zero weakly stable orbit; (b) T2(·) is unitary and weakly stable;

(c) T3 (·) is completely non-unitary and weakly stable. As in the discrete case (see Subsection 11.3.2), we see from the above theorem that a characterisation of weak stability for unitary groups on Hilbert spaces is of special importance. We will discuss more aspects of this problem in Subsection

IV.l.l. At the end of this subsection we present the following classical result describ• ing the part T3(-) in the above theorem if the semigroup consists of isometries, see Foia§, Sz.-Nagy [238, Theorem 111.9.3]. Theorem 4.11 (Wold decomposition). LetT(·) be an isometric C0 -semigroup on a

Hilbert space H. Then H can be decomposed into an orthogonal sum H = H0 g;H1 of T(-)-invariant subspaces such that the restriction ofT(·) to H0 is a unitary semigroup and the restriction ofT(·) to H1 is unitarily equivalent to the unilateral shift on L 2 (.1R+, Y) for a Hilbert spaceY. In addition, dim Y = dim(rg V)_L where Vis the cogenerator ofT(-).

4.3 Characterisation via resolvent In this subsection we discuss a resolvent approach due to Chill, Tomilov [49], see also Eisner, Farkas, Nagel, Sereny [67]. The following main result gives some sufficient conditions for weak stability, see Chill, Tomilov [49] and Eisner, Farkas, Nagel, Sereny [67]. Theorem 4.12. LetT(·) be a C0 -semigroup on a Banach space X with generator

A satisfying so(A) < 0. For x E X and y E X' fixed, consider the following assertions.

(a) (b)

(c)

Joo I(R2 (a +is, A)x, y)l ds da < oo; lo -oo {

1

lim a/

a->0+

00

_ 00

I(R2 (a+is,A)x,y)lds=O;

lim (T(t)x, y) = 0. t->oo

Then (a)::::}(b)::::}(c). In particular, if T(·) is bounded and (a) or (b) holds for all x from a dense subset of X and ally from a dense subset of X', then (T(t))t~o is weakly stable.

4. Weak stability

113

Proof. We first show that (a) implies (b). From the theory of Hardy spaces, see, e.g., Rosenblum, Rovnyak [223, The• orem 5.20], we know that the function f: (0, 1) ~----+ JR+ defined by

f(a) :=

1:

I(R 2 (a+is,A)x,y)ids

is monotone decreasing for a > 0. Assume now that (b) is not true. Then there exists a monotone decreasing null sequence {an}~=l such that (111.28) holds for some c > 0 and all n E N. Take now n, mEN such that an f we have

~

a2'.

By (111.28) and the monotonicity of

This contradicts (a) and the implication (a):::}(b) is proved. It remains to show that (b) implies (c). By (b) we have

1:

I(R2 (a +is, A)x, y)l ds < oo

for every a> 0. Since so(A) ~ 0, Theorem I.3.9 (see also Theorem 1.3.8) implies the inverse Laplace transform representation 1 (T(t)x, y) = 27rt

1

00

-oo

1:

. 2 (a +is, A)x, y) ds e(a+•s)t(R

(III.29)

for all a> 0. We now take t = ~ to obtain

I(T(t)x, y)l

~a

I(R2 (a +is, A)x, y)l ds -t 0

as a-t 0+, sot= ~ -too, proving (c). The last part of the theorem follows from the standard density argument.

D

Remark 4.13. Convergence of the integrals in (b) and hence in (a) in Theorem 4.12 for all x EX andy EX' implies so(A) = w 0 (T) by Corollary 2.19, and hence is not necessary for weak stability of a C0 -semigroup on a Banach space. A useful necessary and sufficient resolvent condition for weak stability is still unknown. In particular, it is not clear whether condition (b) in Theorem b holds for all x and y from dense subsets for weakly stable C0 -semigroups on Banach spaces (and even for unitary groups on Hilbert spaces).

Chapter III. Stability of Co-semigroups

114

5 Almost weak stability In this section we investigate almost weak stability, a concept looking more com• plicated, but being much easier to characterise than weak stability. We will follow Eisner, Farkas, Nagel, Sen)ny [67].

5.1

Characterisation

In this part we always assume T(-) to be a relatively compact semigroup. (This is for example the case if T(·) is bounded and X is reflexive, see Example 1.1.7(a)). Since every weakly stable semigroup is relatively weakly compact, it is not a severe restriction. We begin with a list of equivalent properties motivating our definition of almost weak stability.

Theorem 5.1. Let (T(t))t>o be a relatively weakly compact Co-semigroup on a

Banach space X with generator A. The following assertions are equivalent. -;--=..,....,.----=-,a

(i) 0 E {T(t)x: t 2 0} for every x EX; (i') 0 E {T(t) : t 2 0}

Lu

;

(ii) For every x E X there exists a sequence {t1 }~ 1 C JR+ converging to oo such that limj-too T(tj)x = 0 weakly; (iii) For every x E X there exists a set M C JR+ with density 1 such that lim T(t)x = 0 weakly; t-too, tEM

11t

(iv) lim t-too t (v) (vi)

lim af

a-tO+

I(T(s)x, y)l ds = 0 for all x EX, y EX';

00

_ 00

I(R(a+is,A)x,y)l 2 ds=OforallxEX,yEX';

lim aR(a +is, A)x = 0 for all x EX and s E JR;

a-tO+

(vii) Pa(A)

If,

0

n ilR = 0, i.e., A has no eigenvalues on the imaginary axis.

in addition, X' is separable, then the conditions above are also equivalent to

(ii*) There exists a sequence {t 1 }~ 1 C R+ converging to oo such that limj-too T(tj) = 0 in the weak operator topology; (iii*) There exists a set M C JR+ with density 1 such that

weak operator topology.

lim T(t) t-too, tEM

= 0 in the

Analogously to the discrete case, the (asymptotic) density of a measurable set M c JR+ is

d(M)

:= lim

~-\([0, t] n M),

t--->oo t

5. Almost weak stability

115

whenever the limit exists (here .X is the Lebesgue measure on JR). Note that 1 is the greatest possible density. We will use the following elementary fact, being the continuous analogue of Lemma II.4.2. Lemma 5.2 (Koopman-von Neumann, 1932). Let f: IR+-> IR+ be measurable and

bounded. The following assertions are equivalent. (a) lim -11t f(s) ds = 0; t-+oo t 0

(b) There exists a set M C IR+ with density 1 such that

f(t) = 0. lim t-+oo, tEM

Proof. Assume (b). Then we have for C := supt~o f(t) that -11t f(s)ds=-11t f(s)lM(s)ds+-11t f(s)lMc(s)ds

t

t

0

S

t

0

0

c [0, t]), t1 Jot f(s)lM(s) ds +-(>..(Men

where Me denotes the complement of M in IR+. The first summand on the right• hand side converges to 0 as t -> oo since convergence implies Cesaro convergence, and the second summand converges to 0 as t -> oo since Me has density 0, proving

(a). For the converse implication assume (a) and define for k E N the sets

We have Nk C Nk+l for all k E N. Moreover, since 1Nk(s) < kf(s) holds for every s ~ 0, (a) implies d(Nk) = 0. Therefore there exists an increasing sequence {tk}f= 1 with limk-.oo tk = oo such that (III.30) We show that the set

UNk n [tk-1, tk] 00

M

:=

k=l

has the desired properties. Observe first that fortE [tk-1, tk] n M we have t ~ implies f(t) = 0. lim t-+oo, tEM

Nk, i.e., f(t) S t• which

Chapter III. Stability of C0 -semigroups

116

Moreover, since the sets N'k descrease in k, (III.30) implies that for every k and

t E [tk-1, tk]

So d(M) = 1 and (b) is proved.

0

Proof of Theorem 5.1. The proof of the implication (i') => (i) is trivial. The im• plication (i) => (ii) holds since in Banach spaces weak compactness and weak se• quential compactness coincide by the Eberlein-Smulian theorem (Theorem 1.1.1). If (vii) does not hold, then (ii) cannot be true by the spectral mapping theorem for the point spectrum (see Engel, Nagel [78, Theorem IV.3.7]), hence (ii) =>(vii). The implication (vii) => (i') is the main consequence of the Jacobs-Glicksberg-de Leeuw decomposition (Theorem !.1.19) and follows from the construction in its proof, see Engel, Nagel [78], p. 313. This proves the equivalences (i) 9 (i') 9 (ii) 9 (vii). (vi) 9 (vii): Since the semigroup (T(t))t>o is bounded and mean ergodic by Theo• rem !.2.25, we have by Proposition 1.2.24 that the decomposition X= ker AEBrg A holds and the limit Px := lim aR(a, A)x a--->0+

exists for all x E X with a projection P onto ker A. Therefore, 0 ~ Pa(A) if and only if P = 0. Take now s E R The semigroup (eistT(t))t>O is also rela• tively weakly compact and hence mean ergodic. Repeating the argument for this semigroup we obtain (vi) 9 (vii). -:c=-;-..,.-------c:-7£

(i') => (iii): LetS := {T(t): t 2 0} " ~£(X) which is a compact semitopological semigroup if considered with the usual multiplication and the weak operator topology. By (i) we have 0 E S. Define the operators T(t) : C(S) ----+ C(S) by

(T(t)f)(R)

:=

f(T(t)R),

f

E

C(S), RES.

By Nagel (ed.) [196], Lemma B-II.3.2, (T(t))t>o is a Co-semigroup on C(S). By Example 1.1.7 (c), the set {f(T(t) ·): t 2 0} is relatively weakly compact in C(S) for every f E C(S). It means that every orbit {T(t)f: t 2 0} is relatively weakly compact, and, by Lemma I.1.6, (T(t))t::;:o is a relatively weakly compact semigroup. Denote by

P the mean ergodic projection of (T(t))t>O· Fix(T)

=

n

Fix(T(t))

t?:O

We have

= (1).

Indeed, for f E Fix(T) one has f(T(t)I) = f(I) for all t 2: 0 and therefore f must be constant. Hence Pf is constant for every f E C(S). By definition of the ergodic

5. Almost weak stability

117

projection

(FJ)(O)

= lim

t-+oo

~t }t0 T(s)f(O) ds =

f(O).

(111.31)

R E S.

(111.32)

Thus we have

(Ff)(R)

=

f

f(O) · 1,

E C(S),

Take now x E X. By Theorem 1.1.5 and its proof (see Dunford, Schwartz [63, p. 434]), the weak topology on the orbit {T(t)x: t 2: 0} is metrisable and coincides with the topology induced by some sequence {Yn}~=l C X'\ {0}. Consider fx,n E C(S) defined by RES, !x,n(R) := I(Rx, 11 11 )I,

t:

and

fx

E C (S) defined by

fx(R)

:=

L nEN

By (111.32) we obtain

RES.

11t

11t -

0 = lim t-+oo

1 2n fx,n(R),

t

0

T(s)fx,y(I) ds = lim t-+oo t

Lemma 5.2 applied to the continuous and bounded yields a set M c ~ with density 1 such that lim

t-+oo, tEM

fx(T(t))

0

fx(T(s)) ds.

function~+ 3

t

t-t

f(T(t)I)

= 0.

By definition of fx and by the fact that the weak topology on the orbit is induced by {Yn}~=l we have in particular that lim

t-+oo, tEM

T(t)x = 0 weakly,

proving (iii). (iii) => (iv) follows directly from Lemma 5.2. (iv) => (vii) holds by the spectral mapping theorem for the point spectrum, see Engel, Nagel [78, Theorem IV.3.7]. (iv) {::} (v): Clearly, the semigroup (T(t))t~o is bounded. Take x EX, y EX' and let a> 0. By the Plancherel theorem applied to the function t t-t e-at(T(t)x,y) we have

i:

I

(R(a +is, A)x, y) 12 ds

= 27r

1 00

e- 2 at I (T(t)x, y) 12 dt.

Chapter III. Stability of C0 -semigroups

118

We obtain by the equivalence of Abel and Cesaro limits, see Lemma !.2.23, lim ajoo I(R(a+is,A)x,y)l 2 ds

a--+0+

_

lim a

=

a--+0+

00

lim

t-+oo

Note that for a bounded continuous function we have

1 ( Ct

Jot f2(s) ds )

2

t

2

!t Jo{t I(T(s)x, YW ds.

f : R+

:S (1t Jo f(s) ds

Jo{oo e- ati(T(s)x,y)l

--+

2

ds (III.33)

R+ with C := sup f (R+)

) :S t1 Jot P(s) ds, 2

which together with (III.33) gives the equivalence of (iv) and (v). For the remaining part of the theorem suppose X' to be separable. Then so is X, and we can take dense subsets {Xn f. 0 : n E N} ..) > 0} denoted by Rx (·) and the following assertions are equivalent.

Chapter III. Stability of C0 -semigroups

120 a

.

(1) 0 E {T(t)x : t 2: 0} . (ii) There exists a sequence {tj}~ 1 converging to oo such thatlimj_,oo T(tj)X = 0

weakly.

(iii) There exists a set M C IR+ with density 1 such that

weakly. (iv)

lim

t->oo, tEM

T(t)x = 0

lim~t }ti(T(s)x,y)ids=OforallyEX'. 0

t->oo

(v) lim a1 a->0+

00

_ 00

I(Rx(a+is),y)l 2 ds=OforallyEX 1 •

(vi) lim aRx(a +is)= 0 for all s E R a->0+

(vii) The restriction of A to lin{T(t)x : t 2: 0} has no eigenvalue on the imaginary

axis.

Proof. For the first part of the theorem define Rx(>..) :=

fooo

e->.tT(t)x dt whenever Re >.. > 1.

Denote now by Z the closed linear span of the orbit {T(t)x : t 2: 0}. Then Z is a T(-)-invariant closed subspace of X and we can take a restriction ofT(·) denoted by Tz(-) which is, by Lemma 1.1.6, relatively weakly compact as well. By the uniqueness of the Laplace transform we obtain that R(Az, >..)x = Rx(>..) for every>.. with Re (>..) > 0, where Az denotes the generator of Tz(-). The rest follows from the canonical decomposition X' = Z' EB Z 0 with Z 0 := D {y EX': (z, y) = 0 for all z E Z} and Theorem 5.1.

5.2

Concrete example

We now give a concrete example of an almost weakly stable but not weakly stable C0 -semigroup, again following Eisner, Farkas, Nagel, Sereny [67]. Example 5.9. As in Nagel (ed.) [196], p. 206, we start from a flow on C\{0} with the following properties:

1) The orbits starting in z with

lzl =!= 1 spiral towards the unit circler;

2) 1 is the fixed point of cp and r \ {1} is a homoclinic orbit, i.e., lim 'Pt(z) = lim 'Pt(z)

t->oo

t->-00

= 1 for every z E f.

A concrete example comes from the differential equation in polar coordinates

(r,w) = (r(t),w(t)):

see the following picture.

{~

1- r, 1 + (r 2

-

2rcosw),

5. Almost weak stability

121

Take xo E C with 0 < lxol < 1 and denote by Sx 0 := {rpt(xo) : t 2 0} the orbit starting from x 0 . Then S := Sxo U r is compact for the usual topology of C. We define a multiplication on S as follows. For x = lf't(x0 ) and y = rp 8 (x 0 ) we put

xy := lf't+s(xo). For X E r, X = limn->oo Xn, Xn = lf'tn (xo) E Sxo and y = lf's(xo) E Sxo, we define xy = yx := limn->oo XnY· Note that by ixny- lf's(x)l = ilf's(xn)- lf's(x)i ::S Glxn -xi - - t 0 the definition is correct and satisfies n->ao

xy = rp 8 (x ). For x, y E r we define xy := 1. This multiplication on S is separately continuous and makes S a semi-topological semigroup (see Engel, Nagel [78], Sec. V.2). Consider now the Banach space X:= C(S). By Example I.l.7 (c) the set

{f(s ·) : s E S} c C(S) is relatively weakly compact for every f E C(S). By definition of the multiplication on S this implies that

{! ('Pt (-)) : t 2 0} is relatively weakly compact in C(S). Consider the semigroup induced by the flow, i.e.,

(T(t)f)(x) := f(cpt(x)),

f E C(S), xES.

By the above, each orbit {T(t)f : t 2 0} is relatively weakly compact in C(S) and hence, by Lemma I.1.6, (T(t))t 20 is weakly compact. Note that the strong continuity of (T(t))t>o follows, as shown in Nagel (ed.) [196, Lemma B-II.3.2], from the separate continuity of the flow.

122

Chapter III. Stability of C0 -semigroups

Next, we take Xo := {! E C(S) : f(1) = 0} and identify it with the Banach lattice C 0 (S \ {1} ). Then both subspaces in the decomposition C(S) = X 0 EB (1) are invariant under (T(t))t?O· Denote by (To(t))t?O the semigroup restricted to Xo and by Ao its generator. The semigroup (To(t))t>o remains relatively weakly compact. Since Fix(To) = nt>o Fix(To(t)) = {0}, we have that 0 ~ P""(Ao). Moreover, P""(Ao) n ilR = 0 holds, which implies by Theorem 5.1 that (To(t))t?O is almost weakly stable. To see that (To(t))t?O is not weakly stable it is enough to consider dx 0 E Xb. Since (To(t)f, dx 0 ) = f(rp(t, xo)), f E Xo, f(f) always belongs to the closure of { (To(t)f, 6x 0 ) semigroup (T0 (t))t>o cannot be weakly stable.

:

t > 0} and hence the

We summarise the above as follows. Theorem 5.10. There exist a locally compact space f2 and a positive, relatively weakly compact C0 -semigroup on Co(f2) which is almost weakly but not weakly

stable. The above phenomenon cannot occur for positive semigroups on special Ba• nach lattices; for details and discussion see Chill, Tomilov [50]. Theorem 5.11 (Groh, Neubrander [112, Theorem. 3.2]; Chill, Tomilov [50, The• orem. 7.7]). For a bounded, positive, mean ergodic Co-semigroup (T(t))t?O on a

Banach lattice X with generator (A, D(A)), the following assertions hold.

(i) If X ~ L 1 (f2, ft), then Pcr(A') n ilR = 0 is equivalent to the strong stability

of (T(t))t>O· (ii) If X~ C(K), K compact, then Pcr(A') nilR exponential stability of (T(t))t?O·

= 0 is equivalent to the uniform

Example 5.9 above shows that (ii) does not hold in spaces C0 (f2), f2 locally compact.

6 Category theorems In this section we compare weak and almost weak stability and show that (at least on Hilbert spaces) these two related notions differ drastically. Analogously to the discrete case (see Section II.5), we show following Eisner, Sereny [70] that a "typical" unitary C0-group as well as a "typical" isometric Co-semigroup on a separable Hilbert space is almost weakly but not weakly stable for an appropriate topology. An even stronger result will be shown in Section IV.3.

123

6. Category theorems

6.1

Unitary case

We start from the set ucont of all unitary C0 -groups on a separable infinite• dimensional Hilbert space H. A first step in our construction is the following density result. Proposition 6.1. For every n E N the set of all periodic unitary C0 -groups with

period greater than n is dense in ucont in the norm topology uniform on compact time intervals.

Proof. Take U(·) E ucont and n E N. By the spectral theorem H is isomorphic to L2 (0., 11) for some locally compact space 0. and finite measure 11 and U(·) is unitarily equivalent to a multiplication semigroup U(·) with (U(t)f)(w) = eitq(w) f(w) \:/wE 0., t ~ 0, f for some measurable q : 0. ----. R We approximate the semigroup

qk(w) :=

k27rj

U(·)

-1 (

\:/wE q

as follows. For k

[27rj 27r(j

k'

+

k

L 2 (0., 11)

E

> n define

1)]) , .

J E Z.

The multiplication operator with eitqk (·) we denote by Vk (t), and Vk (·) is a periodic unitary group with period greater than or equal to k > n. Moreover, IIU(t)f- Vk(t)fll

=In leitq(w)- eitqk(w)l llf(w)ll 2

2 dw

47rltl

:S 2ltl s~p lq(w)- qk(w)l·llfll 2 = -k-llfll 2 holds. So limk-+oo IIU(t)- Vk(t)ll = 0 uniformly fort in compact intervals, and the D proposition is proved. Remark 6.2. By a modification of the proof of Proposition 6.1 one can choose the

approximating periodic unitary groups V(-) to have bounded generators. For the second step we need the following lemma. Lemma 6.3. Let H be a separable infinite-dimensional Hilbert space. Then there

exists a sequence {Un(·)}~=l of almost weakly stable unitary groups with bounded generator satisfying limn-+oo IIUn(t) -Ill = 0 uniformly in t in compact intervals. Proof. By isomorphy of all separable infinite-dimensional Hilbert spaces we can assume that H = L 2 (JR) with respect to the Lebesgue measure. Taken EN and define Un(·) on L 2 (JR) by itq(s)

(Un(t)f)(s) := e-n-f(s),

S E

2

lR, f E L (JR),

where q: lR----. [0, 1] is a strictly monotone increasing function.

124

Chapter III. Stability of C0 -semigroups Then all Un(-) are almost weakly stable by Theorem 5.1 and we have

IIUn(t)- Ill =sup le_n_ itq(s)

sEIR

- 11 ::;

[fort::; nn] ::;

2t

len - 11 ::; it

n

--->

0,

n---> oo, 0

uniformly for t in compact intervals. We consider the topology on the space ucont coming from the metric

d(U(·), V(·)) :=

L

sup [ ]II U(t)x - V(t)x ·II tE -n,n . J J n,j=l 2J llxj II 00

for U V E ucont

'

'

where {xj}~ 1 is some fixed dense subset of H \ {0}. Note that this topology corresponds to the strong convergence uniform on compact time intervals in IR and is a continuous analogue of the strong*-topology for operators used in Subsection II.5.1. We further denote by S{Aont the set of all weakly stable unitary groups on H and by W{Aont the set all almost weakly stable unitary groups on H. The following result shows density of W{Aont in ucont. Proposition 6.4. The set Wuont of all almost weakly stable unitary groups with

bounded generator is dense in ucont.

Proof By Proposition 6.1 it is enough to approximate periodic unitary groups by almost weakly unitary groups. Let U(·) be a periodic unitary group with period T. Take E > 0, n EN, x1, ... , Xn E H \ {0} and to> 0. We have to find an almost weakly stable unitary group V(·) with IIU(t)xj - V(t)xj II ::; E for all j = 1, ... , n and all t with ltl ::; to. By Engel, Nagel [78, Theorem IV.2.26] we have .L ( H-_ EBkEZker A- -2nik) , 7-

where A denotes the generator of U(·). So we can assume without loss of generality that { Xj }j= 1 is an orthonormal system of eigenvalues of A. Define now the T(-)-invariant subspace Ho := lin{x1, ... , Xn} and B :=A on H0 . Further, since H is separable, the decomposition

holds, where dimHk = dimH0 for every k E .N. For a fixed orthonormal basis

{ej}j= 1 of each Hk we define Bej := Bxj and extend B to a bounded linear operator on H. From the construction follows that

) EB .L ... EB .L ker ( B - -2ni.An) H = ker ( B - -2ni.Al 7 7

,

125

6. Category theorems

· th e e1genva . lue of A (and there£ore of B) correspond"mg t o th e wh ere 21ri>.j IS 7 eigenvector Xj. Let Xj := ker (B- 21r;>. 1 ) for every j = 1, ... , n. On every Xj the operator

B is equal to 21r;-'J I. Note further that all Xj are infinite-dimensional. By Lemma 6.3 for every j there exists an almost weakly stable unitary group TJ (·) on XJ 27rti>..j

such that IITJ(t) - e ,. Ill < c for every t with Jtl :::; to. Denote now by T(·) the orthogonal sum of TJ(·) which is a weakly stable unitary group with bounded generator. Moreover, we obtain that

for every t with Jtl :::; t 0 and the proposition is proved.

D

We now prove a category theorem for weakly and almost weakly stable uni• tary groups, being analogous to its discrete counterpart given in Subsection II.5.1. Theorem 6.5. The set S[iont of weakly stable unitary groups is of first category and

the set W[iont of almost weakly stable unitary groups is residual in ucont. Proof. We first prove that Suont is of first category in llxll = 1 and consider

Mt

:= {

U(-)

E ucont:

I(U(t)x, x)l :::;

ucont.

Fix

X E

H with

1}.

Note that all sets Mt are closed. For every weakly stable U(·) E ucont there exists t > 0 such that U EMs for all s ~ t, i.e., U(·) E Nt := ns?_tMt. So we obtain

Suont

c

UNt.

(III.34)

t>O

Since all Nt are closed, it remains to show that ucont \ Nt is dense for every t. Fix t > 0 and let U(·) be a periodic unitary group. Then U(·) ~ Ms for some s 2: t and therefore U(·) ~ Nt. Since, by Proposition 6.1, periodic unitary groups are dense in ucont, the set Suont is of first category. To show that Wuont is residual we take a dense subspace D = { Xj }~ 1 of H and define Wjkt

:= {

U(·)

E ucont:

I (U(t)xj, Xj) I
O Wjkt are also open.

00

W[/ont

=

j,k=l

WJk·

(III.35)

Chapter III. Stability of Co-semigroups

126

The inclusion "c" follows from the definition of almost weak stability. To prove the converse inclusion we take U(·) ~ Wziont and t > 0. Then there exists x E H with llxl/ = 1 and r.p E lR such that U(t)x = eit 0, which implies I (U(t)x, x) I = 1. Take Xj ED with llxj- xll ::::; ~- Then we have

I(U(t)xj, Xj) I= I (U(t)(x- Xj), x- Xj) + (U(t)x, x)- (U(t)x, x- Xj) 2

- (U(t)(x- Xj),x) I~ 1-llx- Xjll - 2llx- Xjll >

1 3·

So U(·) ~ Wj 3 , hence U(·) ~ n_rk=l Wjk, and equality (III.35) holds. Therefore Wziont is residual as a dense countable intersection of open sets. 0

6.2 Isometric case In this subsection we consider the space Icont of all isometric C0 -semigroups on H endowed with the strong topology uniform on compact time intervals and prove category results as in the previous subsection. We again assume H to be separable and infinite-dimensional. Note that Icont is a complete metric space with respect to the metric given by the formula

d(T(·), S(·)) :=

L

00

n,j=l

sup [

tE O,n

J

IIT(t)x · - S(t)x ·II .

J

2J llxj I

J

forT(·), S(·) E Icont,

where {xj}~ 1 is a fixed dense subset of H \ {0}. We further denote by SI:ont the set of all weakly stable, and by WI:ont the set of all almost weakly stable isometric Co-semigroups on H. The main tool is the classical Wold decomposition from Theorem 4.11. In addition, we need the following easy lemma (see also Peller [211]).

Lemma 6.6. Let Y be a Hilbert space and let R( ·) be the right shift semigroup on H := l2 (N, Y). Then there exists a sequence {Un(-)}~=l of periodic unitary groups on H converging strongly to R( ·) uniformly on compact time intervals.

Proof. For every n E N we define Un (t) by

(Un(t)f)(s)

:=

{

s ~ n; Rn(t)f(s), s E [0, n],

f(s),

where Rn(·) denotes the n-periodic right shift on the space £ 2 ([0, n], Y). Then every UnO is a C0 -semigroup on L 2 (JR+, Y) which is isometric and n-periodic, and therefore unitary.

6. Category theorems Fix

f

127

E L 2 (IR+, Y) and T

IIUn(t)f- R(t)fll 2 =

> 0. Then fort::; T and n > T we have

1 llf(s)00

::1

00

=

f(s

+ n- t)ff 2 ds +fat llf(s + n- t)ff 2 ds

llf(s)ff 2 ds +

1:t 2/:T

2/:t llf(s)ff ds:::::: 2

llf(s)ll 2 ds +

[:t

llf(s)ll 2 ds

llf(s)ff 2 ds n~

0

uniformly on [0, T], and the lemma is proved.

0

As a consequence we obtain the following density result for periodic Co• groups in Icont.

Proposition 6.7. The set of all periodic unitary C0 -groups is dense in Icont.

Proof. Let V(·) be an isometric semigroup on H. Then by Theorem 4.11 the or• thogonal decomposition H = H0 E9 H 1 holds, where the restriction Vo(·) of V(·) to H 0 is unitary, H 1 is unitarily equivalent to L 2 (IR+, Y) for some Y and the restric• tion V1(·)of V(·) on H 1 corresponds to the right shift semigroup on L 2 (IR+, Y). By Proposition 6.1 and Lemma 6.6 and their proofs we can approximate both semi• groups Vo (·) and V1(·) by unitary periodic ones with period in N and the assertion follows. 0 Also the following density result for almost weakly stable semigroups is a consequence of Wold's decomposition and the results of the previous subsection.

Proposition 6.8. The set is dense in I cant.

W:Iont

of almost weakly stable isometric C0 -semigroups

Proof. Let V(·) be an isometric Co-semigroup on H, Ho and H1 the orthogonal subspaces from Theorem 4.11 and V0 (-), V1(-) the corresponding restrictions of V(·). By Lemma 6.6, the semigroup V1(·) can be approximated by unitary Co• groups on H 1. The assertion now follows from Proposition 6.4. 0 Using the same idea as in the proofs of Theorem 6.5, Propositions 6.7 and 6.8, we obtain the following category theorem for weakly and almost weakly stable isometric C0 -semigroups.

Theorem 6.9. The set S:Iont of all weakly stable isometric Co-semigroups is of first category and the set Wyant of all almost weakly stable isometric C0 -semigroups is residual in I cant.

6.3 Contractive case: remarks It is not clear whether one can prove results analogous to Theorems 6.5 and 6.9 for contractive C0-semigroups (as in the discrete case in Section II.5.3). We point out the main difficulty.

128

Chapter III. Stability of C0 -semigroups

Let ccont denote the set of all contraction semigroups on H endowed with the metric for T, S E ccont, where {xj}~ 1 is a fixed dense subset of H \ {0}. The corresponding convergence is the weak convergence of semigroups uniform on compact time intervals. Note that this metric is a continuous analogue of the metric used in Section II.5.3. The following example, see Eisner, Sereny [71], shows that the space ccont is not complete (or compact) in general, and hence the Baire category theorem cannot be applied as it was done in the proofs of Theorems 6.5 and 6.9. Example 6.10. Consider X:= [P, 1 ::; p

< oo, and the operators An defined by

An(Xl, · · ·, Xn, Xn+l, · .. , X2n, · · .)

:= (xn+l, Xn+2, ... , X2n, Xl, X2, ... , Xn, X2n+l, X2n+2, ... )

exchanging the first n coordinates of a vector with its next n coordinates. Then IIAnll :S: 1 implies that An generates a Co-semigroup Tn(-) satisfying IITn(t)ll :S: et for every n E N and t ~ 0. -2 The operators An converge weakly to zero as n--) oo. Moreover, An= I for every n E N. Therefore -

oo

Tn(t) =

=

L

tk

Ak

k! n =

k=O

et - e-t -

2

L

00

An

k=O

+

et

t2k+ 1

-

(2k + 1)!An +

+ e-t 2

u

I ~

t2k

oo

L

k=O

et

(2k)!I

+ e-t 2

I

#-

I,

where "a" denotes the weak operator topology, and this convergence is uniform on compact time intervals. However, the limit does not satisfy the semigroup law. By rescaling An := An- I we obtain a sequence of contractive semigroups converging weakly and uniformly on compact time intervals to a family which is not a semigroup while the bounded generators converge weakly to -I (which is a generator). Thus it is not clear whether

ccont

is a Baire space.

Note finally that related residuality results appear in the context of positive semigroups, see Bartoszek and Kuna [17] for recent category results for Markov semigroups on the Schatten class C1 and Lasota, Myjak [160] for an analogous result for stochastic semigroups. Remark 6.11. In Section IV.3 we improve the above results using the notion of

rigidity.

7. Stability via Lyapunov's equation

129

7 Stability via Lyapunov's equation In the last section of this chapter we characterise stability and boundedness of C0 -semigroups on Hilbert spaces via a Lyapunov equation. This goes back to Lyapunov [166] in 1892 where he investigated uniform exponential stability of matrix semigroups. As in the discrete case, we use a positivity approach as, e.g., Nagel (ed.) [196, Section D-IV.2], Nagel, Rhandi [199], Groh, Neubrander [112], Batty, Robinson [28] and Alber [4, 5]. A different approach is in Guo, Zwart [113], see also Datko [55]. In what follows, only the characterisation of uniform exponential stability is classical, while the other results are recent. Our main tools are implemented semigroups as defined in Subsection I.4.4.

7.1

Uniform exponential stability

We start with the infinite-dimensional version of Lyapunov's result and charac• terise uniform exponential stability. Theorem 7.1. LetT(·) be a C0 -semigroup on a Hilbert space H with generator A

and T(·) be the corresponding implemented semigroup with generator A. Then the following assertions are equivalent.

(i) T( ·) is uniformly exponentially stable. (ii) T (·) is uniformly exponentially stable. (iii) 0 E p(A) and 0 ::; R(O, A). (iv) There exists 0::; Q E .C(H) with Q(D(A))

c D(A*) such that

AQ = A*Q + QA =-I on D(A).

(III.36)

(v) There exists 0::; Q E .C(H) with Q(D(A)) c D(A*) such that AQ = A*Q + QA::; -I on D(A). In this case, the solution Q of (III.36) is unique and given as Q = R(O, A)I = 00 T(t)*T(t) dt, where the integral is defined with respect to the strong operator topology.

J0

Equation (III.36) is called Lyapunov equation.

Proof. (i)¢:?(ii) follows directly from Lemma I.4.8(a), and (ii)¢:?(iii) by Theorem I.4.9. We now show (iii)=?(iv). By Theorem I.4.9, (iii) implies that T(-) is uniformly exponentially stable and the representation R(O, A)X x = 000 T(t)X x dt holds for every X E .C(H) and x E H. Define Q E £(H) by

f

Q := R(O,A)I = -A- 1 I

E

D(A)

Chapter III. Stability of C0 -semigroups

130

which is positive semidefinite and satisfies

Qx = R(O, A)Ix =

1oo T(t)*T(t)xdt

for every x E H. Moreover, we have by the definition of A that A*Q AQ =-I, and (iv) follows. (iv):::}(v) is trivial, and it remains to show that (v):::}(i). Assume (v). We have by (v)

+ QA

=

lot T(s)I ds :S-lot T(s)AQds = Q- T(t)Q :S Q, where all integrals are defined strongly in .C(H). Since the family

J

(J~ T(s)I ds)

t>O

is monotone increasing in .C(H), the integral 0 T(s)I ds converges for the we~k operator topology, cf. Lemma !.4.1 (b). In particular, the integral 00

converges for every x E H, and therefore T (·) is uniformly exponentially stable by 0 Datko's theorem, see Theorem 2.10.

7.2 Strong stability The idea to characterise also strong stability via Lyapunov equations appears in Guo, Zwart [113]. We give an alternative proof.

Theorem 7.2. LetT(·) be a Co-semigroup with generator A on a Hilbert space H, and letT(·) be its implemented semigroup with generator A. Then the following assertions are equivalent.

(i) T(·) is strongly stable. (ii) limt_. 00 (T(t)Sx,x) = 0 for all S E .C(H) and x E H.

(iii) For every a > 0 there exist (unique) 0 :S Qa, Qa D(A*) and Oa(D(A*)) C D(A) satisfying

E

.C(H) with Qa(D(A))

C

(A- ai)*Qa + Qa(A- ai) =-I on D(A), (A- ai)Qa + Qa(A- ai)* =-I on D(A*) such that lim a(Qax, x) = 0

a-+0+

and

sup a(Qax, x) a>O

< oo for all x E H.

(III.37)

7. Stability via Lyapunov's equation

131

In this case, Qa

= R(a, A)I = 1oo e-atT(t)*T(t) dt,

(III.38)

Qa = R(a, A*)I = 1oo e-atT(t)T(t)* dt,

(III.39)

where the integrals are defined strongly. Proof. (ii)=>(i) follows directly from (T(t)Ix,x) = IIT(t)xll 2 • (i)=>(ii): Let x E H and S E £(H)sa· Then there is a constant c > 0 such

that -cl ~ S ~ cl. This implies I(T(t)Sx,x)l ~ ci(T(t)Ix,x)i = ci!T(t)xll 2 . By (i) we thus have limn--->oo (T(t)Sx, x) = 0 for every S E £(H) sa and hence for every S E £(H). We prove (i)#(iii) simultaneously. Assume that (i) or (iii) holds. From The• orem 7.1 applied to the semigroups (e-atT(t))t~o we know that the rescaled Lyapunov equations in (iii) are satisfied by (the unique) Qa := R(a,A)I = J000 e-atT*(t)T(t) dt, where the integral is defined strongly on H. Therefore we have

(Qax, x) =

1 00

e-atiiT(t)xll 2 dt.

Analogously, Qa := R(a, A*)I and

(Qax, x) = 1

00

e-atiiT(t)*xll 2 dt.

The condition Qa(D(A*)) C D(A) follows from D(A**) = D(A) and Theorem 7.1 (iv). By Lemma !.2.6 we have that (III.37) is equivalent to

11t

lim t--->oo t

0

11t

IIT(s)*xll 2 ds < oo IIT(s)xll 2 ds = 0 and supt>O t o

for all x E H. By Theorem 3.7 for p = q = 2 this is equivalent to strong stability D ofT(·), and the proof is finished.

Remark 7.3. By the above proof, one can replace "=" by Lyapunov equations in (iii).

"~"

in the rescaled

7.3 Boundedness We now characterise boundedness of Co-semigroups on Hilbert spaces via Lya• punov equations, see Guo, Zwart [113] for the result and a different proof.

Theorem 7.4. LetT(-) be a Co-semigroup on a Hilbert space H. Then the following assertions are equivalent.

(i) T(·) is bounded.

132

Chapter III. Stability of C0 -semigroups

(ii) For every a > 0 there exist (unique) 0 :S Qa, Qa D(A*) and Qa(D(A*)) C D(A) such that

E

£(H) with Qa(D(A))

C

(A- ai)*Qa + Qa(A- ai) =-I on D(A), (A- ai)Qa + Qa(A- ai)* =-I on D(A*) with

sup a(Qax, x)

a>O

< oo

and

sup a(Qax, x)

a>O

< oo '1/x

E

H.

(III.40)

In this case, Qa and Qa satisfy representations (III.38) and (III.39), respectively, where the integrals are defined strongly.

The proof is analogous to the one of Theorem 7.2 using Theorem 1.6 instead of Theorem 3.7. Moreover, one can replace "=-I" by ":S -I" in both equalities in (ii) as well. Moreover, one can replace (III.40) by the condition used in Guo, Zwart [113] supaiiQall < oo and supaiiQall < oo

a>O

a>O

which is equivalent to (III.40) by the polarisation identity and the uniform bound• edness principle.

Final remark As in the discrete case, see Section II.6, weak stability is much more delicate to treat. The main problem occurs again when T(-) is a unitary group. Open question 7.5. Is it possible to characterise weak stability of C0 -semigroups on Hilbert spaces via some kind of Lyapunov equations?

Chapter IV

Connections to ergodic and measure theory Ergodic theory has motivated important notions and results in operator theory such as, e.g., the mean ergodic theorem. In this chapter we connect stability of operators and C0 -semigroups back to analogous notions in harmonic analysis and ergodic theory. In the last section we describe "typical" asymptotics of discrete and continuous semigroups using rigidity, a notion well-known in these two areas.

1 Stability and the Rajchman property In this section we show parallels between operator theory and measure theory. It turns out that weak stability of operators and Co-semigroups has a direct analogue in measure theory called the Rajchman property. Rajchman measures are certain measures on the unit circle or on the real line (or, more generally, on a locally compact abelian group). Starting his investigation in 1922, Rajchman was motivated by questions on the uniqueness of trigonometric series. Since then many people have worked on Rajchman measures (see, e.g., Rajchman [216, 217], Milicer-Gruzewska [185], Lyons [174, 175, 176, 177, 178], Bluhm [34], Koerner [151], Goldstein [98, 99], Ransford [218]), and we refer to Lyons [179] for a historical overview.

1.1

From stability to the Rajchman property and back

We now introduce the Rajchman property for measures and relate it to weak stability of unitary operators and semigroups.

Chapter IV. Connections to ergodic and measure theory

134

Discrete case We first consider an example which will help us to understand the general situation.

Example 1.1. Let p, be some probability measure on the unit circle r and consider the multiplication operator Ton the space H := L 2 (f, p,) defined by

(TJ)(z)

:=

zf(z),

z E f.

Then T is unitary, hence not strongly stable, and we are interested in its weak stability. Since z E Pu(T) if and only if p,( {z}) > 0, the Jacobs-Glicksberg-de Leeuw decomposition (Theorem 1.1.15) implies the following equivalence. Tis almost weakly stable{=:} p, is a continuous measure, i.e.,

p,({z})

= 0 for all z E f.

To characterise weak stability, we first observe that

for every f E H, where we identify r with [0, 2n]. Therefore, if Tis weakly stable and for f = 1, we obtain that the Fourier coefficients P.n of p, must satisfy {27r

Mn := Jo

eins dp,(s)----> 0 as n----> oo.

(IV.l)

Conversely, assume that condition (IV.1) holds. Take f,g defined by f(s) := eims and g(s) := eils and observe that

(Tn f,g) =

1 2

7r

ei(n+m-l)s dp,(s)----> 0 as n----> oo.

By the standard linearity and density argument we obtain limn-+oo(Tn f, g) = 0 for every f,g E L 2 (f,p,), i.e., Tis weakly stable. Note that a unitary operator is weakly stable if and only if its inverse is, hence in condition (IV.l) we can also taken----> -oo. This and Theorem II.4.1 prove the following proposition (see Lyons [175] for the first equivalence in (a)).

Proposition 1.2. For the operator T defined above, the following holds. (a) T is weakly stable {=:} limn-+oo P,n

= 0 {=:} limlnl-+oo Mn = 0.

(b) T is almost weakly stable {=:} p, is continuous {=:} limj--+oo Mni = 0 for some nj ----> oo {=:} limj--+oo Mni = 0 for some {nj}f= 1 with density 1.

1. Stability and the Rajchman property

135

Remark 1.3. The fact that a measure J.t is continuous if and only if its Fourier coefficients converge to zero on a subsequence of N (or, equivalently, of Z) with density 1 also follows from Wiener's lemma, see e.g. Katznelson [146, p. 42], and Lemma 11.4.2.

Measures satisfying (a) in the above proposition were given their own name.

Definition 1.4. A Radon measure on r is called Rajchman if its Fourier coefficients converge to zero. Proposition 1.2 states that the operator T is weakly stable if and only if the measure J.t is Rajchman. For the properties of Rajchman measures we refer to Lyons [175, 179]. We only remark that every absolutely continuous measure is Rajchman by the Rie• mann-Lebesgue lemma and every Rajchman measure is continuous by Wiener's lemma, but none of the converse implications holds. The first example of a singu• lar Rajchman measure was a modified Cantor-Lebesgue measure constructed by Menshov [184] in 1916. Not absolutely continuous Rajchman measures with re• markable properties have been constructed later, see, e.g., Bluhm [34] and Korner [151]. On the other side, the classical Cantor-Lebesgue middle-third measure is an example of a continuous measure which is not Rajchman. By our considerations above, each continuous non-Rajchman measure J.t on r induces an almost weakly but not weakly stable unitary operator on LP(f, J.t). By the spectral theorem (see e.g. Conway [52, Theorem IX.4.6]), one can reduce the general situation to the previous example. Indeed, consider a contraction Ton a Hilbert space H. By Theorem 111.4.7, its restriction T1 to the subspace H1 := { x : IITnxll = IITmxll = llxll Vn E N} is unitary and its restriction to H is weakly stable. So T is weakly stable if and only if T1 is weakly stable. We now apply the spectral theorem to the unitary operator T1 and obtain, for each x E H 1 , a measure J.tx on r such that the restriction ofT to lin{Tnx : n = 0, 1, 2, ... } is isomorphic to the multiplication operator Mzf(z) := zf(z) on L 2 (r, J.tx)· So we are in the context of Example 1.1 and see that

r

lim Tnx = 0 weakly

n-->oc

{::::=:}

J.tx is Rajchman.

Thus we have the following characterisation of stability of unitary operators via spectral measures.

Proposition 1.5. Let H be a Hilbert space with orthonormal basis S and U be a unitary opemtor on H. Then U is weakly stable if and only if the spectml measures J.tx are Rajchman for every xES. Moreover, U is almost weakly stable if and only if J.-tx is continuous for every x E S. This gives a measure theoretic approach to weak stability of operators. How• ever, since there is no simple characterisation of Rajchman measures, this approach is difficult to use in concrete situations.

Chapter IV. Connections to ergodic and measure theory

136

Continuous case We now relate weak stability of Co-semigroups to the Rajchman property of mea• sures on the real line. The results are analogous to the discrete case discussed in the previous subsection. We again begin with an example which is typical for the general situation.

Example 1.6. Let J.L be a probability measure on L2(~, J.L), take the multiplication operator

(Af)(s)

:=

isf(s),

and, on the space H .•

~

s E ~'

with its maximal domain D(A) := {f E H : g E H for g(s) := isf(s)}. The C 0 -group T(·) generated by A is

(T(t)f)(s)

:=

eist f(s),

s, t

E ~'

f

E

H.

It is unitary and hence not strongly stable, but we can ask for weak or almost weak stability. Note that is E PO'(A) if and only if J.L( {s}) > 0 and the Jacobs-Glicksberg-de Leeuw decomposition (Theorem 1.1.19) implies

T(·) is almost weakly

stable~

J.L is continuous, i.e., J.L( {s}) = 0 Vs

E

R

On the other hand, we see that

holds for every

f

1:

E H. Therefore, if T(·) is weakly stable, then

FJ.L(t)

:=

eist dJ.L(s)---+ 0 as t---+ oo,

(IV.2)

where :FJ.L denotes the Fourier transform of f.L· Conversely, if (IV.2) holds, then limt-. 00 (T(t)f, f) = 0 for every f having constant absolute value. Since the linear span of { ein·} ~=-oo is dense in H and T(·) is contractive, limt--+oo(T(t)f, f)= 0 for every f E H, so by the polarisation identity T(·) is weakly stable. Note further that a unitary group is weakly stable fort---+ +oo if and only if it is weakly stable fort---+ -oo. This proves the following proposition (see Lyons [175]).

Proposition 1.7. The following assertions hold for the above semigroup T(-). (a) T(·) is weakly stable~ limt--+oo:FJ.L(t)

=0

~ limJtJ--+ooFJ.L(t)

= 0.

(b) T(·) is almost weakly stable~ J.L is continuous~ limj--+oo :FJ.L(tJ) = 0 for some itJ I ---+ oo ~ limJtJ--+oo, tEM :FJ.L( t) = 0 for some M C ~ with density

1.

1. Stability and the Rajchman property

137

As before, a finite measure on IR is called Rajchman if its Fourier transform converges to zero at infinity. We refer to Lyons [175, 179] and Goldstein [98, 99] for a brief overview on Rajchman measures on the real line and their properties (the second author uses the term "Riemann-Lebesgue measures"). We just mention that, as in the dis• crete case, absolutely continuous measures are always Rajchman by the Riemann• Lebesgue lemma, and all Rajchman measures are continuous by Wiener's theorem. However, there are continuous measures which are not Rajchman and Rajchman measures which are not absolutely continuous (see Lyons [179] and Goldstein [99]). It is now a consequence of the considerations above that each continuous non• Rajchman measure gives rise to an almost weakly but not weakly stable unitary Co-group. For a concrete example of a unitary group with bounded generator whose spectral measures are not Rajchman see Engel, Nagel [78, p. 316]. Finally, we can connect Rajchman measures on IR to Rajchman measures on r (as defined in Example 1.1). Indeed, for a probability measure J.L on IR take its image v under the map s f---t eis. Then J.L is Rajchman if and only if v is so. Using the spectral theorem for unitary operators on Hilbert spaces, questions concerning weak (and almost weak) stability of C0 -semigroups can be reduced to the previous example. Indeed, consider an arbitrary contraction semigroup T(-) on a Hilbert space H. By Theorem III.4.7 the restriction TI(·) ofT(·) to the subspace W := {x : IIT(t)xll = IIT*(t)xll = llxll Vt 2: 0} is unitary and the restriction to Wl_ is weakly stable. In order to check weak stability, it remains to investigate the unitary (semi)group T1 (-). Applying the spectral theorem to A1 we obtain for each x E H 1 a measure J.Lx on IR such that the restriction of A1 to lin{T(t)x: t 2: 0} is isomorphic to the multiplication operator Misf(s) := isf(s) on L 2 (IR, J.Lx)· By Example 1.6, we see that lim T(t)x = 0 weakly {::::=:} J.Lx is Rajchman. t---.oo

Note further that by Theorem 5.1 T(·) is almost weakly stable if and only if J.lx is continuous for every x. So we have the following. Proposition 1.8. A unitary C0 -group U(·) on a Hilbert space H with orthonormal basis S is weakly stable if and only if J.Lx is Rajchman for every xES. Moreover, U (·) is almost weakly stable if and only if J.Lx is continuous for every x E S.

1.2

Category result, discrete case

Inspired by the connection of stability and the Rajchman property, one can ask whether results analogous to the category results for operators and C0 -semigroups (see Sections II.5 and III.6) hold for measures, i.e., whether a "typical" Radon measure is continuous and not Rajchman for some appropriate topology. In fact, a much stronger fact holds, see also Nadkarni [193, Chapter 7].

Chapter IV. Connections to ergodic and measure theory

138

Denote by M the set of all Radon probability measures on r. On this set we consider the weak* topology, i.e., limn->..)

!1 dp,

instead of

Moreover, we may assume that the intervals [sj, Sj +a] (respectively,

[1 - a, 1]) are disjoint. Define now Vj := J.I({=J}) l[sj,sJ+a]A and v :=Meant+ I:j Vj. Then vis con• tinuous and v([0,1]) = Mcont([0,1]) + I:jM({sj}) = p,([0,1]) = 1. Moreover, for

1. Stability and the Rajchman property l

139

= 1, ... , n we have

< c LJL({sj}):::; £, j

0

and the proposition is proved.

We now introduce a class of measures having a much stronger property than being non-Rajchman. Definition 1.11. A probability measure J.l on the unit circle f is called rigid if its Fourier coefficients satisfy limj--+oo P.ni = 1 for some subsequence {nj }~ 1 C N. Analogously, J.l is called >.-rigid for some ).. E f if there exists an increasing sequence {ni }~ 1 c N such that limj--+oo P.ni = >-.. We finally call a measure f-rigid if it is >-.-rigid for every ).. E f.

Note that, J.l being a probability measure, one always has

IP.nl :::;

1 for all

n E Z, so >-.-rigidity is a kind of extremal property.

Remark 1.12.

1) For any ).. E f, >-.-rigidity implies rigidity. This follows from 1 E {)..n : n E N} and the fact that the limit sets are closed.

2) If ).. is irrational, i.e., satisfying ).. i. ei7l"IQ, then every >-.-rigid measure is automatically f-rigid, since {).. n : n E N} is dense in f. 3) As a converse to 1), there is an elegant argument how to produce >-.-rigid (and hence f-rigid if).. is irrational) measures from a given rigid one via a rescaling argument, see Nadkarni [193, p. 50]. The easiest example of rigid measures are point measures. Indeed, a point measure 8>. is always rigid, and it is f-rigid if and only if).. is irrational. Examples of singular rigid measures will be briefly discussed in Subsection 2.2. We are now ready to prove the following category result due to Choksi, Nadkarni [51], see also Nadkarni [193, Chapter 7]. Theorem 1.13. The set of all continuous rigid measures is residual in M. In

particular, the set MR of all Rajchman measures is of first category, while the set Meant of all continuous measures is residual. Proof. We start by proving that the set Consider the sets

Mk,n

Mrig

of all rigid measures is residual.

= {J.l: IP.n- 11 < ~},

where P.n is the n-th Fourier coefficient of J.l· These sets are open for the weak* topology by the definition of Fourier coefficients. Define furthermore the sets

Nk,l := Un'2:lMk,n = {J.l: IP.n- 11
2/ d we obtain that 11 ~ Wjk,

therefore 11 ~ n_rk=l Wk and (IV.3) is proved. Thus Wcont is a dense countable intersection of open sets by the above and Proposition 1.10. This proves the assertion. 0 Analogously to the first part of the above proof and using a delicate rescaling argument mentioned in Remark 1.12, one can prove that for,\ E r, all ,\-rigid mea• sures are residual in Mas well, see Nadkarni [193, 7.17]. Using this and Remark 1.12 2), we obtain the following result.

Theorem 1.14. The set of measures 11 satisfying

(a) 11 is continuous, i.e., limj_, 00 fln 1 = 0 for some {nJ}

C Z

with density 1,

2. Stability and mixing

141

(b) f is contained in the limit set of the Fourier coefficients of JL is residual in M for the weak* topology. Remark 1.15. One can add several other interesting properties to the above list of "typical" properties for measures. For example, the measures which are orthogonal to a given one are residual, which implies in particular that singular measures are residual. For a detailed discussion of this subject we refer to Nadkarni [193, Chapter 7].

1.3 Category result, continuous case Analogous phenomena hold for probability measures on the real line. By MIR we denote the set of all probability measures JL on JR. endowed with the weak* topology, i.e., JLn ----7 JL if and only if (!, JLn) ----7 (!, JL) for every f E Cb(JR.). Since MIR is a closed subset of the unit ball in (Cb(JR.))' for the weak* topology, it is a complete metric (and compact) space. By just the same arguments as in the discrete case, we obtain the following.

Theorem 1.16. The set of measures JL satisfying

(a) JL is continuous, (b) JL is f-rigid, i.e., f is contained in the limit set of the Fourier transform of JL

is residual in MIR for the weak* topology. In particular, the set of all Rajchman measures on JR. is of first category, while the set of all continuous measures is residual. In Example 3.21 b) below we briefly discuss abstract examples of such mea• sures coming from weakly mixing f-rigid automorphisms.

2 Stability and mixing We now turn our attention to the analogue of stability in ergodic theory. Ergodic theory deals with a probability space (D, ~. JL) and a measure pre• serving transformation (m.p.t.) 'P : n ----7 n, i.e., a measurable map satisfying JL('P- 1 (B)) = JL(B) for all B E ~. The continuous analogue is a measure pre• serving semiflow (m.p. semiflow) ('Pt)t;:::o, where each 'Pt is a m.p.t., the function (w, t) H 'Pt(w) is measurable and 'Po = I d, 'Pt+s = 'Pt'Ps holds for all t, s 2:: 0. We refer to Cornfeld, Fomin, Sinai [53], Krengel [154], Petersen [212], Halmos [118], or Tao [240, Chapter II] for basic facts and further information. To translate the above concepts into operator theoretic language, fix 1 ~ p < oo and define X:= V(D, E,JL) and T E £(X) by Tf := f o!p, f E V(D,JL). Then T is a linear operator on X which is isometric since, 'P being measure preserving,

142

Chapter IV. Connections to ergodic and measure theory

Thus, for cp invertible and p = 2, we obtain a unitary operator T. Note that the operator T has relatively weakly compact orbits for p > 1 by the theorem of Banach-Alaoglu and for p = 1 by Example 1.1.7 (b) with u = 1. Analogously, for a m.p. semiflow ('Pt) define (T(t))t~o by T(t)f := f o 'Pt, f E LP(f!,tJ). Then (T(t))t~o is a semigroup of linear isometries which is strongly continuous by Krengel [154], §1.6, Thm. 6.13. Thus, if one/each 'Pt is invertible (i.e., we start with a flow) and p = 2, then T(·) becomes a unitary C 0 -group. Moreover, T(·) is relatively weakly compact (for p = 1 use again Example 1.1.7 (b) with u = 1). We call T the induced operator and T(·) the induced C0 -semigroup. This oper• ator (and each operator of the C0 -semigroup) has remarkable additional properties: it is multiplicative on the Banach algebra L 00 (f!,tJ) and positive on each of the Banach lattices LP(f!, tJ), i.e., it maps positive functions into positive functions. This implies, as an easy consequence of the Perron-Frobenius theory for such op• erators (see Schaefer [227, Section V.4]), that the boundary spectrum a(T) n r as well as the boundary point spectrum Pa(T) n r are cyclic subsets of r, i.e., A E a(T) n f implies An E a(T) n f for all n E Z, and analogously for Pa(T).

2.1 Ergodicity We begin with ergodicity, the basic notion in ergodic theory. Discrete case

For am.p.t. cp on (f!,~,tJ), a set BE~ is called cp-invariant if tJ(cp- 1 (B)L::.B) = 0, and cp is ergodic if for every cp-invariant set B either tJ(B) = 0 or tJ(B) = 1 holds. In other words, ergodic transformations leave no non-trivial set invariant. Translating this property into a property of the induced operator leads to the following. Theorem 2.1. A m.p.t. cp is ergodic if and only if the induced operator satisfies

FixT = (1). In this case every eigenvalue ofT is simple.

Proof. Note first that the constant functions are fixed under T. Assume Fix T = (1). If B E ~ is cp-invariant, we obtain T1B = 1B, hence 1B = 1 or 1B = 0 by assumption, and thus cp is ergodic. Assume now that cp is ergodic and take f E FixT. Since each set Br := {w E f! : Ref(w) > r}, r E lR, is cp-invariant,

it must be trivial, hence Re f is constant a.e .. Analogously, Im f is constant and thus f E (1). The first part of the theorem is proved. To show the second part take A E Pa(T) and 0 =/= f EX with Tf = f o cp = Af, Then Tf ="X ·f and, since T is isometric, A E f and therefore Tlfl = lfl. Thus If I is a non-zero constant function by Fix T = (1). Assume If I = 1 and take another eigenfunction g corresponding to the eigenvalue A with lgl = 1. Then 0 T(gf) = gf implies gf = 1, so f = g and A is simple.

2. Stability and mixing

143

By the mean ergodic theorem applied to the induced operator T, ergodicity can be reformulated into an asymptotic property of the transformation r.p and the operator T.

Corollary 2.2. For a m.p.t. r.p on (D, I:, 11), 1 ~ p < oo, and the induced operator T on X = £P(f2, 11), the following assertions are equivalent.

(i) (ii)

'{J

is ergodic.

N 1 lim - N """"'1-l('P-n(B) ~ + 1 n=O N-+oo N

n C)= 11(B)11(C) for every B, C E I:.

r

1 LTnf= fdl-l·lforeveryfEX. (iii) lim - N Jn + 1 n=O N-+oo

Proof. (i):::}(iii) As mentioned above, T has relatively weakly compact orbits and is therefore mean ergodic by Theorem 1.2.9. If Pis the mean ergodic projection ofT and f E X, we obtain by ergodicity of r.p that P f = Cl for some constant C by Proposition 1.2.8. To determine this constant observe that

where we used that '{J preserves the measure 11· This implies C = claim follows. (iii):::}(ii) For B, C E I: and f := lB, we have by (iii),

N

N1-l('P-n(B) n C)= N 1+ 1 LNl

+1 L

1

n=O C

n=O

__. L i lB

d/1

lcp-n(B)

d/1 = N

fn fd/1,

Nl

+1 L

1

n=O C

and the

Tn f d/1

1 d/1 = !1(B)/1(C)

as N---> oo. To show (ii):::}(i), take B with 11('P- 1 (B)6.B) = 0. Property (ii) implies

11(B) 2 Hence either 11(B)

1

N

lim - N L = N-+oo +1

n=O

1-l('P-n(B) n B)= 1-l(B).

= 0 or 11(B) = 1 showing ergodicity of r.p.

D

It follows from the proof that it suffices to take C = B in (ii).

Remark 2.3. One can interpret (iii) as "time mean equals space mean" which is a version of the famous ergodic hypothesis by Boltzmann from around 1880 (see, e.g., Halmos [118, Introduction] and [68, Chapter 1]).

Chapter IV. Connections to ergodic and measure theory

144 Continuous case

Ergodicity for semiflows is defined analogously, and we state the relevant properties without proof, see, e.g., Cornfeld, Fomin, Sinai [53, §1.2, 1.4]. A measure preserving semiflow ( cpt )t>o is called ergodic if the only invariant sets under all cpt are trivial, i.e., if f-L(cp;1 (B)6B) = 0 for every t 2 0 implies either f-l(B) = 0 or f-l(B) = 1. Again, ergodicity can be characterised in terms of the induced C 0-semigroup T(·). Theorem 2.4. Let cp be a measure preserving semifiow and T(·) the induced C0 semigroup with generator A on LP(O, J.l), 1 ::; p < oo. Then cp is ergodic if and only ifFixT(·) := nt>o FixT(t) = ker A= (1). In this case every eigenvalue of A

is simple.

-

Analogously to the discrete case, mean ergodicity ofT(·) allows the following characterisation of ergodicity of (cpt)t?:O· Corollary 2.5. For a measure preserving semifiow (cpt )t?:O and the induced semi• group T(-) on LP(O, J.l), 1 ::; p < oo, the following assertions are equivalent. (i) (cpt)t?:O is ergodic.

11t

(ii) lim t-+oo

(iii) lim

t-+oo

t

~ t

0

f-l(cp; 1 (B)

n C)ds = J.l(B)f-l(C) for every B, C

Jot T(s)fds = Jn{ fdf-l· 1 for every f

E

I:.

E LP(O, f-l).

Again one can take C =Bin (ii). We see that, in the discrete and in the continuous case, ergodicity is equivalent to an asymptotic property of cp expressed by assertions (ii) and (iii) in Corollaries 2.2 and 2.5. Property (ii) means that cp; 1 (A) becomes "equidistributed" in the Cesaro sense. Stronger asymptotic properties are discussed below.

2.2 Strong and weak mixing The so-called "mixing properties" in ergodic theory are closely related to weak and almost weak stability as studied in Sections 11.4 and 11!.5. We explain this relation now. Discrete case A measure preserving transformation cp on a probability space (0, I:, f-l) is called strongly mixing if

2. Stability and mixing

145

for any two measurable sets B, C E E. The transformation cp is called weakly mixing if N 1 lim - N IJL(cp-n(B) n C)- JL(B)JL(C)I = 0

+1

N-->oo

I: n=O

holds for all B, C. These concepts play an important role in ergodic theory, and we refer to Cornfeld, Fomin, Sinai [53], Krengel [154], Petersen [212], Walters [254] or Halmos [118] for further information. Strong mixing implies weak mixing and weak mixing implies ergodicity by Corollary 2.2, but the converse implications do not hold in general. A famous result of Halmos [116] and Rohlin [221] states that a "typical" m.p.t. (for the strong operator topology for the induced operators) is weakly but not strongly mixing. However, explicit examples of weakly but not strongly mixing transformations are not easy to construct, see e.g. Lind [171] for a concrete example and Petersen [212, p. 209] for a method of constructing such transformations. An important class of weakly but not strongly mixing transformations are so-called rigid transformations discussed briefly in Example 3.12 b) below. We first translate the above concepts into the operator-theoretic language (see, e.g., Halmos [118, pp. 37-38]). Proposition 2.6. Let cp be a m.p.t. on a probability measure space (D, E, JL), 1 :s; p < oo, and let T be the induced linear operator on X := LP(D, JL) defined by T f := f o cp. Denote further by P the projection given by P f ·- f 0 f dfL · 1, f E X. Then the following assertions hold.

(a) cp is strongly mixing if and only if lim (Tnf,g) = (Pf,g)

n-->oo

for all f EX, g EX'.

(b) cp is weakly mixing if and only if 1 N lim --'""" I(Tnf,g)- (Pf,g)l = 0 N--->oo N + 1 L..,;

for all f EX, g EX'.

n=O

Proof. (a) By definition, cp is strongly mixing if and only if lim

rTnlB . lc dfL rlB dfL . rlc dfL

n-->oo } !1

=

} !1

} !1

for every B, C E :E, which, by a standard density argument in the Banach space X, is equivalent to

146

Chapter IV. Connections to ergodic and measure theory

for every f EX and g EX'. (b) Analogously to the above, cp is weakly mixing if and only if

1

J~oo N + 1

N

L

I(Tnf,g)- (Pf,g)l

n=O

1

N

= J~ N + 1 L I((Tn- P)f,g)l = 0 n=O

holds for every f := ls and g := lc with B, C E I:. By the triangle inequality, this is equivalent to the same property for every f, g in the linear hull of characteristic functions and, by the standard density argument, for every f EX and g EX'. 0 In order to relate mixing to stability properties, i.e., convergence to 0, we only need to eliminate the constant functions which are always fixed under T. To this end consider the decomposition X = (1) EB X 0 , where

Xo := { f E X :

L

f dp =

0}

is closed and T-invariant. Note that the restriction T0 := Tlxo has relatively weakly compact orbits as well. The following connects mixing properties of cp to stability properties of T 0 and follows directly from the above proposition.

Corollary 2.7. For a m.p.t. cp on a probability space (D, I:, p) and the corresponding operator To on Xo C LP(D, p), 1 :::; p < oo, the following assertions hold.

(a) cp is strongly mixing if and only if T0 is weakly stable. (b) cp is weakly mixing if and only if T0 is almost weakly stable. Remark 2.8. Every weakly but not strongly mixing transformation induces an almost weakly but not weakly stable operator. However, as mentioned above, an explicit example of such a transformation and hence of such an operator is not easy to construct, see e.g. Petersen [212, Section 4.5], Example 3.12 b) and Example 3.17 below. Continuous case We now give the continuous analogues of the above concepts and present some results connecting mixing flows to stable C 0-semigroups. A measure preserving semiflow ('Pt )t;::o on a probability space (n, I:, J-L) is called strongly mixing (or just mixing) if lim p(cpf:

t-+oo

1

(B)

n C)= p(B)p(C)

holds for every B, C E I:. The semiflow ('Pt)t;::o is called weakly mixing if for all B,C E I: we have

11t

lim -

t-->oo

t

0

IJ-L(cp:; 1 (B) n C)- p(B)p(C)I ds = 0.

2. Stability and mixing

147

Strong mixing implies weak mixing and weak mixing implies ergodicity, but the converse implications do not hold in general. For an example of a weakly mixing flow which is not mixing see, e.g., Cornfeld, Fomin, Sinai [53, Chapter 14], Katok [141], Katok, Stepin [144], Avila, Forni [11], Ulcigrai [244], Scheglov [228], Kulaga [159] and also Example 3.21 b) below. Note that there seems to be no continuous analogue of Halmos' and Rohlin's result on "typicality" of weakly but not strongly mixing flows. The following continuous analogue of Proposition 2.6 holds. Proposition 2.9. Let ('Pt )t>o be a measurable measure preserving semiflow on a probability measure space (0, I;, J.L), 1 :S p < oo, and letT(·) be the induced Co•

semigroup on X = LP(O, p,). Let further P be the projection given by P f ·• f dJ.L · 1, f EX. Then the following assertions hold.

J0

(a) ('Pt )t>o is strongly mixing if and only if lim (T(t)f,g) = (Pf,g)

t-+oo

for all f EX, g EX'.

(b) ('Pt k::o is weakly mixing if and only if

11t

lim-

t-+oo

t

0

I(T(s)f,g)-(Pf,g)ids=O

for all f EX, g EX'.

As before, the constant functions are fixed under the induced Co-semigroup. So we again consider the decomposition X = (1) EBXo for the closed and (T(t) )t2:o• invariant subspace

We denote the restriction of (T(t))t>o to X 0 by (To(t))t>o and its generator by A 0 . The semigroup (To(t))t2:0 remains relatively weakly compact. This leads to the following characterisation of mixing of ('Pt )t2:0 in terms of stability of the semigroup (To(t))t2:0· Corollary 2.10. For a measure preserving semiflow ('Pt )t>o on a probability space

(0, I;, J.L), 1 :S p < oo, and the corresponding semigroup To(·) on Xo C LP(O, J.L), the following assertions hold.

(a) ('Pt )t>O is strongly mixing if and only if To (·) is weakly stable. (b) ('Pt )t>o is weakly mixing if and only if T0 ( ·) is almost weakly stable. So every weakly but not strongly mixing flow yields an almost weakly but not weakly stable C0-semigroup.

Chapter IV. Connections to ergodic and measure theory

148

3 Rigidity phenomena Inspired by results in ergodic and measure theory, we now describe the "typical" (in the Baire category sense) asymptotic behaviour of unitary, isometric and con• tractive operators on separable Hilbert spaces using the notion of rigidity. We will see that the "typical" behaviour is quite counterintuitive and can be viewed as "random": The strong limit set of the powers of a "typical" contraction contains the unit circle times the identity operator, while most of the powers converge weakly to zero. We also give the continuous analogue for unitary and isometric C0 -(semi)groups. This in particular generalises category results from Sections 11.4 and 111.5.

3.1

Preliminaries

We first define rigid and f-rigid operators and C0 -semigroups and make some elementary observations.

Definition 3.1. A bounded operator T on a Banach space X is called rigid if strong- lim TnJ J---+00

=I

for some subsequence { nj} ~ 1 C N.

Note that in the above definition one can remove the assumption limj-+oo nj = oo. (Indeed, if n1 does not converge to oo, then rna = I for some no implying rnno = I for every n E N.)

Remark 3.2. Rigid operators have no non-trivial weakly stable orbit. In particular, by the Foia§-Sz.-Nagy decomposition (see Theorem 11.3.9), rigid contractions on Hilbert spaces are necessarily unitary. As trivial examples of rigid operators take T := >..I for IA.I = 1. Moreover, arbitrary (countable) combinations of such operators are rigid as well, as the fol• lowing proposition shows.

Proposition 3.3. Let X be a separable Banach space and let T E .C(X) be power

bounded with discrete spectrum, i.e., satisfying

H

= lin{x EX:

Tx

= A.x

for some A. E f}.

Then T is rigid. Proof. Since X is separable, the strong operator topology is metrisable on bounded sets of .C(H) (take for example the metric d(T,S) := I:;o IITzj- Szjll/(2jllzjll) for a dense sequence { z1}~ 1 c X \ {0}). So it suffices to show that I belongs to the strong closure of {Tn}nEN· ForE> 0, mEN and XI, ... ,Xm EX, we have to find n E N such that IITnxj - Xj II < E for every j = 1, ... , m. Assume first that each Xj is an eigenvector with llxj II = 1 corresponding to some unimodular eigenvalue Aj, and hence IITnxj- Xjll = IA.j -1lllxll· Consider

3. Rigidity phenomena

149

the compact group rm and the rotation

q, ... , Am)· By a classical recurrence theorem, see e.g. Furstenberg [92, Theorem 1.2], there exists n such that lr.pn(l)- 11 < E, i.e., l.\j- 11

_ (We used the fact that for irrational.\ the set pn};;c'= 1 is dense in r and that limit sets are always closed.) The simplest examples are again operators of the form .\I, l.\1 = 1. Indeed, T = AI is .\-rigid, and it is r -rigid if and only if.\ is irrational. Analogously, one can define rigidity for strongly continuous semigroups.

Definition 3.6. A C0 -semigroup (T(t)t>o) on a Banach space is called .\-rigid for ), E r if there exists a sequence {t1 }~ 1 C JR+ with limj--+oo tj = oo such that strong- lim T(t 1 ) =.\I. J--+00 Semigroups which are 1-rigid are called rigid, and if they are rigid for every.\ E r they are called r -rigid.

Chapter IV. Connections to ergodic and measure theory

150

As in the discrete case, >.-rigidity for some for some irrational >. is equivalent to r-rigidity.

>. implies rigidity, and >.-rigidity

Remark 3. 7. Again, rigid semigroups have no non-zero weakly stable orbit. This implies by the Foia§-Sz.-Nagy decomposition, see Theorem III.4.7, that every rigid semigroup on a Hilbert space is automatically unitary.

The simplest examples of rigid C0 -semigroups are given by T(t) = eiat I for some a ERIn this case, T(·) is automatically r-rigid whenever a=/= 0. Moreover, one has the following continuous analogue of Proposition 3.3. Proposition 3.8. Let X be a separable Banach space and let T(·) be a bounded

C0 -semigroup with discrete spectrum, i.e., satisfying H = lin{x EX: T(t)x = eitax for some a E IR and all t 2: 0}. Then T (·) is rigid. So rigidity becomes non-trivial for operators (C0 -semigroups) having no point spectrum on the unit circle. Recall that for operators with relatively compact orbits, the absence of point spectrum on r is equivalent to almost weak stability, i.e.,

weak- lim Tnj = 0 for some subsequence {nJ} with density 1, J-+00

and analogously for C0 -semigroups, see Theorems II.4.1 and III.5.1. Restricting ourselves to Hilbert spaces, we will see that almost weakly stable and r-rigid operators and semigroups are the rule and not just an exception.

3.2 Discrete case: powers of operators In this section we describe the "typical" (in the Baire category sense) asymptotic behaviour of unitary, isometric and contractive operators on separable Hilbert spaces. As in Section II.5, we take a separable infinite-dimensional Hilbert space H and denote by U the set of all unitary operators on H endowed with the strong* operator topology. The set of all isometric operators on H with the strong operator topology is denoted by I, and C will be the space of all contractions on H with the weak operator topology. All these spaces are complete metric and hence Baire. We now prove the first but basic step to describe the asymptotics of contrac• tions. Theorem 3.9. Let H be a separable infinite-dimensional Hilbert space. The set

M := {T: lim Tnj =I strongly for some nj

--+

oo}

J-+00

is residual for the weak operator topology in the set C of all contractions on H. This set is also residual for the strong operator topology in the set I of all isometries and in the set U of all unitary operators for the strong* operator topology.

3. Rigidity phenomena

151

Proof. We begin with the isometric case.

Let {xt}~ 1 be a dense subset of H\ {0}. Since one can remove the assumption limj--+oo nj = oo from the definition of M, we have

M

= {T E I:

:l{nj}_~ 1 C N with lim

J--+00

Tnixz

= xz

'ill EN}.

(IV.4)

Consider the sets

M, := { T E I:

~ 2'll~riiiiT•x,- xdl oo Tm = U strongly, hence limm->oo T;:; = un strongly for every n EN. However, Tm E Mk means that

Since r;:; converges strongly and hence weakly to un for every n, and hence the lth summand on the left-hand side of the above inequality is dominated by 21 ~ 1 which form a sequence in l 1 ' we obtain by letting m ----7 00 that

contradicting the periodicity of U.

D

We now show that one can replace I in Theorem 3.9 by >.I for any>. E r. To this purpose we need the following modification of Proposition 11.5.1. Lemma 3.10. Let H be a Hilbert space, >. E

r

unitary operators U with un = >.I for some n unitary operators for the norm topology.

and N E N. Then the set of all ~ N is dense in the set of all

Proof. Let U be a unitary operator, >. = eia E r, N E N and E > 0. By the spectral theorem U is unitarily equivalent to a multiplication operator U on some L 2 (0,JL) with (U f)(w) = cp(w)f(w), Vw E 0, for some measurable cp: 0----+ r := {z E C :- izl = 1}. We now approximate the operator U as follows. Take n I1 - e--;:;-27ril :::; E and define for O:j := e i( n: +27rj) --;:;- , j = 0, ... , n,

>

N such that

0

1/J(w) := O:j_ 1 , Vw

E cp- 1 ({z E

The multiplication operator

r:

arg(o:j_ 1 ):::; arg(z)

P corresponding to 'ljJ

satisfies

< arg(o:j)}).

pn = eia.

Moreover,

IIU- Fll =sup jcp(w)- '1/J(w)l:::; E wErl

proving the assertion.

D

We now describe the "typical" asymptotic behaviour of contractions (iso• metries, unitary operators) on separable Hilbert spaces. For an alternative proof in the unitary case based on the spectral theorem and an analogous result for measures on r see Nadkarni [193, Chapter 7]. Theorem 3.11. Let H be a separable infinite-dimensional Hilbert space. Then the

set of all operators T satisfying the properties

3. Rigidity phenomena (1) there exists {n1 } ~ 1

153

c N with density 1 such that lim Tni = 0

J---+00

(2) for every >. E

r

weakly,

there exists {nj>-)} ~ 1 with limj-+oo nY) = oo such that (.\)

lim Tni

J---+00

= >.I

strongly

is residual for the weak operator topology in the set C of all contractions. This set is also residual for the strong operator topology in the set I of all isometries as well as for the strong* operator topology in the set U of all unitary operators. Recall that every contraction satisfying (2) is unitary, cf. Remark 3.2.

Proof. By Theorems 11.5.4, 11.5.9, 11.5.12, operators satisfying (1) are residual in C, I and U. We now show that for a fixed >. E r, the set M of all operators T satisfying limj-+oo Tni = >.I strongly for some sequence {n1 } ~ 1 is residual. We again prove this first for isometries and the strong operator topology. Take a dense set {xz}l= 1 of H \ {0} and observe that

M = {T E I: :J{nj} with lim Tnixz = Axz Vl EN}. J---+00

We see that M = n~ 1 Mk for the sets

Mk

:=

{

T

EI :

00

1 1 ~ 21 llxziii1Tnxz - Axzll < k

for some n }

which are open for the strong operator topology. Therefore M is a Gs-set which is dense by Lemma 3.10, and the residuality of M follows. The unitary case goes analogously, and we now prove the more delicate con• traction case. To do so we first show that

M = {T E C: :J{nj}f= 1 with lim (Tnixz,xz) = >-llxzll 2 Vl EN}. J---+00

As in the proof of Theorem 3.9, the non-trivial inclusion follows from

II(Tni- >.I)xll 2 = 11Tnixll 2 - 2Re (Tnix, Ax)+ llxll 2 ~ 2Re(llxll 2

For the sets

-

(Tnix,>.x)) = 2Re(:\((>.J -Tni)x,x)).

Chapter IV. Connections to ergodic and measure theory

154

we have the equality M = n%"= 1 Mk. Note again that it is not clear whether the sets Mk are open for the weak operator topology, so we use another argument to show that the complements Mk are nowhere dense. By Lemma 3.10 it suffices to show that Mk n U;. = 0 for the complement Mk and the set U;. of all unitary operators U satisfying un =AI for some n EN. This can be shown as in the proof of Theorem 3.9 by replacing I by AI. Since A.-rigidity for an irrational .X implies f-rigidity, the proof is complete.

D We now present basic constructions leading to examples of operators with properties described in Theorem 3.11. Example 3.12. a) A large class of abstract examples of f-rigid unitary operators which are almost weakly stable comes from harmonic analysis, more precisely, from f-rigid measures, see Subsection 1.1. Recall that A.-rigid continuous measures form a dense G5 (and hence a residual) set in the space of all probability measures with respect to the weak* topology, see Theorem 1.14. Take a A.-rigid measure J.L for some irrational .X. The unitary operator given by (U f)(z) := zf(z) on L 2 (f, J.L) satisfies conditions (1) and (2) of Theorem 3.11. To show (2), it again suffices to prove that U is A.-rigid since .X is irrational. By our assumption on J.L, there exists a subsequence {nj} ~ 1 such that the Fourier

f

coefficients of J.L satisfy limj_, 00 Mnj = limj_, 00 027r einj 8 dJ.L( s) = .X, where we again identify f with [0,21!']. Thus, limj-;oo(Unjf,f) = .XII/11 2 for every f E L2 (f,J.L) with absolute value 1. By

we see that limj_,oo Unj f = A.f for every character, and therefore for every f E L 2 (f, J.L) by the density argument, implying that U is f-rigid. Generally, a unitary operator U on a separable Hilbert space satisfies condi• tions (1) and (2) of Theorem 3.11, i.e., is f-rigid and almost weakly stable if and only if every spectral measure of U is continuous and f-rigid. The "if" direction follows by arguments as in the proof of Proposition 3.3. b) Another class of examples of rigid almost weakly stable unitary operators comes from ergodic theory. A measure preserving transformation 'P on a prob• ability space (n, E, J.L) is called rigid if there exists a subsequence {nj} ~ 1 such that lim J.L(A61{J-nj (A))= 0 for every A E E. J->00

Consider now the induced operator Ton H := L 2 (0,J.L) defined by (Tf)(w) := f('P(w)). By

3. Rigidity phenomena

155

and since characteristic functions span £ 2 (0, J.L), we see that Tis rigid if and only if the transformation

oo ani = .X for the above sequence {nj}· Nadkarni [193, pp. 4950] showed that the set of all a such that the limit set of {ani} ~ 1 contains an irrational number has full Lebesgue measure in r. Every such a leads to a f-rigid operator aT. We now show that one cannot replace the operators AI in Theorem 3.11 by any other operator.

Proposition 3.14. Let V E £(H) be such that the set

Mv := {T:

3{nj}~ 1

such that .lim Tni = V strongly} J--->00

is dense in one of the spaces U, I or C. Then V is a multiple of identity. Proof. Consider the contraction case and assume that the set of all contractions T such that weak-limj--->oo Tni = V for some {nj}~ 1 is dense in C. Since every such operator T commutes with V by TV= limj--->oo Tni+ 1 = VT, we obtain by assumption that V commutes with every contraction. In particular, V commutes with every one-dimensional projection implying that V =.XI for some .X E C.

156

Chapter IV. Connections to ergodic and measure theory

The same argument works for the spaces I and U using the density of unitary 0 operators in the set of all contractions for the weak operator topology.

Remark 3.15. In the above proposition, one has V = >.I for some ).. E r in the unitary and isometric case. Moreover, the same holds in the contraction case if Mv is residual. (This follows from the fact that the set of all non-unitary contractions is of first category inC by Remark 3.2 and Theorem 3.9, see also Theorem V.l.22 below. Remark 3.16. It is not clear whether Theorem 3.11 remains valid under the addi• tional requirement

{>. E C:

1>-.1

< 1} ·I C {Tn:

n E

NV,

(IV.6)

where a denotes the weak operator topology. Since countable intersections of resid• ual sets are residual and the right-hand side of (IV.6) is closed, this question takes the following form: Is, for a fixed).. with 0 < 1>-.1 < 1, the set M>. of all contractions T satisfying >.IE {Tn : n EN} residual? Note that each M>. is dense inC since, for ).. = reis, it contains the set (7

{cU : 0 < c < 1, en = r and un = eis I for some n} which is dense in C by the density of unitary operators and a natural modification of Lemma 3.10. We finally mention that absence of rigidity does not imply weak stability, or, equivalently, absence of weak stability does not imply rigidity, as shown by the following example.

Example 3.17. There exist unitary operators T with no non-trivial weakly stable orbit which are nowhere rigid, i.e., such that limj---+oo Tni x = x for some subse• quence {nj}~ 1 implies x = 0. (Note that such operators are automatically almost weakly stable by Proposition 3.3 and the Jacobs-Glicksberg-de Leeuw decompo• sition.) A class of such examples comes from mildly mixing transformations which are not strongly mixing, see Furstenberg, Weiss [93], Frat,;zek, Lemanczyk [90] and Frat,;zek, Lemanczyk, Lesigne [91].

3.3

Continuous case: C0-semigroups

We now give the continuous analogue of the above results for unitary and isometric strongly continuous (semi)groups. Let H be again a separable infinite-dimensional Hilbert space. We recall the objects introduced in Section III.6. We denote by ucont the set of all unitary Co• groups on H endowed with the topology of strong convergence of semigroups and their adjoints uniformly on compact time intervals. This is a complete metric and

3. Rigidity phenomena

157

hence a Baire space for

d(U(·), V(·)) := ~ SUPtE[-n,n]IIU(t)xj- V(t)xjll ~ 2JIIxjll n,j=l where {xj}~ 1 is a fixed dense subset of H \ {0}. We further denote by ycant the set of all isometric Co-semigroups on H endowed with the topology of strong convergence uniform on compact time intervals. Again, this is a complete metric space for

d(T(·), S(·)) :=

L 00

n,j=l

sup [ JIIT(t)x - S(t)x ·II tE O,n . J J 2J llxj II

forT(·), S(·) E ycant_

The proofs of the following results are similar to the discrete case, but require some additional technical details. Theorem 3.18. Let H be a separable infinite-dimensional Hilbert space. The set

Meant:= {T(·): lim T(tj) =I strongly for some tj __, oo} J->00

is residual in the set ycont of all isometric Co-semigroups for the topology cor• responding to strong convergence uniform on compact time intervals in ~+. The same holds for unitary Co-groups and the topology of strong convergence uniform on compact time intervals in ~Proof. We begin with the unitary case.

Choose {xz}b 1 as a dense subset of H\ {0}. Since one can replace limj....,oo tj = oo in the definition of Meant by {tj }~ 1 C [1, oo) we have

Meant= {T(·) E ucant: :J{tj} E [1, oo):

lim T(tj)xz = xz 'v'l EN}.

(IV.7)

J->00

Consider now the open sets

and Mkont := Ut~l Mk,t· We have

n 00

Mcont =

Mkont,

k=l

and hence Mcont is a G8 -set. Since periodic unitary C0 -groups are dense in ucont by Proposition III.6.1, and since they are contained in Meant, we see that Mcont is residual as a countable intersection of dense open sets. The same arguments and the density of periodic unitary operators in ycont (see Proposition III.6.7) imply the assertion in the isometric case. 0

Chapter IV. Connections to ergodic and measure theory

158

The following continuous analogue of Lemma 3.10 allows us to replace I by >.I. Lemma 3.19. Let H be a Hilbert space and fix A E r and N EN. Then for every unitary Co-group U(·) there exists a sequence {Un(-)}~= 1 of unitary C0 -groups

such that

(a) For every n EN there exists T 2: N with Un(T) =>.I, (b) limn_, 00 IIUn(t)- U(t)ll = 0 uniformly on compact intervals in R

Proof. Let U(·) be a unitary C0-group on H, A= eia E r. By the spectral theorem, H is isomorphic to L 2 (0, p,) for some finite measure space (0, p,) and U(·) is unitarily equivalent to a multiplication group U(·) given by (U(t)J)(w) = eitq(w) f(w),

wEn,

for some measurable q : n - t R To approximate U(·), let N EN, c > 0, to> 0 and take m 2: N, mEN, such that Ill- e 2;:';; II -:5. c/(2to). Define for O!j := e{;;.+~), j = 0, ... , m,

p(w)

:= aj_ 1 for all wE cp- 1 ({z E r: arg(aj_t) -:5. arg(z)

< arg(aj)}).

The multiplication group V(-) defined by V(t)f(w) := eitp(w)f(w) satisfies V(m) = eia. Moreover,

IIU(t)f- V(t)fll 2

=In leitq(w)- eitp(w)l llf(w)ll 2

-:5. 2ltl sup lq(w)- p(w)lllfll 2 wEf!

2

< cllfll 2

uniformly in t E [-to, to].

0

We now obtain the following characterisation of the "typical" asymptotic behaviour of isometric and unitary Co-(semi)groups on separable Hilbert spaces. Theorem 3.20. Let H be a separable infinite-dimensional Hilbert space. Then the

set of all 0 0 -semigroups T(·) on H satisfying the following properties

(1) there exists a set M

C

R+ with density 1 such that lim

t->oo,tEM

T(t) = 0 weakly,

(2) for every A E r there exists {tJA)}~ 1 with limj_,oo t)A) _lim

J->00

T(tfl) = )..[

strongly

= oo

such that

3. Rigidity phenomena

159

is residual in the set of all isometric Co -semigroups for the topology of strong convergence uniform on compact time intervals in IR+. The same holds for unitary

Co-groups for the topology of strong convergence uniform on compact time intervals inR Proof. By Theorems III.6.5 and III.6.9, Co-(semi)groups satisfying (1) are residual in Icont and ucont. We show, for a fixed..\ E f, the residuality of the set M(>.) of all C 0 -semigroups T(·) satisfying strong-limj-+oo T(tj) =AI for some sequence {ti }~ 1 converging to infinity. We prove this property for the space Icont of all isometric semigroups and the strong operator convergence uniform on compact intervals, the unitary case goes analogously. Take..\ E f and observe M(>.)

= {T(·) E ront:

3{tj} C [1, oo) with lim T(tj)Xl J-+00

= ..\x1 'Vl

EN}

for a fixed dense sequence {x1}~ 1 C H \ {0}. Consider now the open sets

M

·=

k,t.

{T(·)

E

ront.

.

~ II(T(t)- ..\I)x!ll ~} r5 one has IIR(.\, T)ll S 1~ for some M and every >. tJ_ L:w. Theorem 1.4. (See Haase [114, Prop. 3.1.1 and 3.1.15]) LetT be a bounded sectorial

operator with dense range. Then T can be embedded into an analytic Co-semigroup. Since T is embeddable if and only if cT is embeddable for any 0 -=/= c E C (consider Tc(t) := rtei 1, n EN.

(V.7)

(iii) V-I is injective and has dense range; there exists J-Lo > 1 such that (1, J-Lo) E p(V) and

I [(I- V)R(J-L, V)t I :S

2nM

(J-L + 1)n

for all J-L E (1, J-Lo), n EN.

(V.8)

174

Chapter V. Discrete vs. continuous

In all these characterisations the resolvent R(A, V) of the cogenerator had to be used. In fact, the boundedness ofT(·) is not equivalent to the power bounded• ness of V. In order to prove some results in this direction we first need a technical lemma on Laguerre polynomials whose proof (in a more general case) can be found in [74]. Lemma 2.3. Let L~ (t) denote the first generalised Laguerre polynomial, i.e.,

Ll(t)= n

~ ~

m=O

(-l)m ( n+1 m! n-m

)tm.

(V.9)

Then the inequality

(V.lO) holds for some constants C1, C2 and all n EN. We are now ready to prove the result of Brenner, Thomee [41] giving the upper bound for the growth of the powers of the cogenerator of a bounded C 0 semigroup. We follow the proof in [7 4]. Theorem 2.4. LetT(-) be a bounded Co-semigroup on a Banach space with cogen•

erator V. Then IIVnll ::; Cyln some C and every n EN. Furthermore, this estimate cannot be improved, i.e., there exists a Banach space and a bounded C0 -semigroup such that IIVnll 2:: Cyln for some C > 0 and every n EN. Proof. Using Lemma 1.3.6 and boundedness of the semigroup we observe that IIVn II ::; 1 +2M

1 IL~-1 00

(2t) I e-tdt

(V.ll)

forM := supt>o IIT(t)ll and all n E N. Lemma 2.3 immediately implies IIVnll ::; E N.

C yin for some -C and every n

We now show that this estimate is sharp. Consider X:= C0 ([0, oo)) and the left shift semigroup T(·) on X given by (T(t)J) (s) = f(t + s). Note that T(·) is contractive and the powers of its cogenerator V are given by the formula 00

(Vn f) (s) = f(s) - 21 L;_ 1(2t)e-t f(t + s )dt by Lemma 1.3.6. Define now h(s) := sign(L~_ 1 (2s)). We obtain formally that

and hence

I (Vnh) (s)l 2:: Cy'n for every s 2:: 0

2. Cogenerators

175

by Lemma 2.3. It remains to approximate h by continuous functions. Thus, for every c > 0 there exists a function !c E Co([O, oo)) with IIIII = 1 such that the 0 above estimate holds within an error of at most c, which finishes the proof.

Remark 2.5. By the standard renorming procedure (see Lemma III.1.4), one de• rives from the above an example of a contractive Co-semigroup on a Banach space such that the powers of its cogenerator grow as .fii. This shows that the Foia§• Sz.-Nagy theorem (see Subsection 1.3.1) fails in Banach spaces. For more examples see Subsection 2.3 below. On Hilbert spaces the following question is still open.

Open question 2.6. LetT(·) be a C0 -semigroup on a Hilbert space with cogenerator V. Is boundedness ofT(·) equivalent to power boundedness of V? Guo, Zwart [113] gave a partial answer to this question under additional assumptions.

Theorem 2.7 (Guo, Zwart). LetT(·) be a Co-semigroup on a Hilbert space with generator A and cogenerator V. Then the following assertions hold.

(a) If T (·) is bounded and A - 1 generates a bounded C0 -semigroup as well, then V is power bounded. (b) If T(·) is analytic, then its boundedness is equivalent to power boundedness ofV. We refer to Guo, Zwart [113] for the proofs and further results. It is not clear whether assertion (a) in the above theorem holds for semigroups on Banach spaces as well.

2.2

Characterisation via cogenerators of the rescaled semigroups

In this subsection we characterise boundedness of a Co-semigroup using the co• generators of the rescaled semigroups. We begin with the following observation. If A generates a contractive or bounded C0 -semigroup, then all operators T A, T > 0, do so as well. However, as we will see in Subsection 2.3, it is not always true that the operators

Vr := (rA + I)(rA- I)- 1 ,

T

> 0,

(V.12)

remain contractive when V is, so one needs another condition to obtain equiva• lence.

Proposition 2.8. For a densely defined operator A on a Banach space X, the fol• lowing assertions are equivalent.

(i) A generates a contraction Co-semigroup on X.

176

Chapter V. Discrete vs. continuous

(ii) (O,oo) C p(A) and the operators Vr satisfy

IIVr - Ill :S 2 (iii) There exists To

>0

such that

for all T

(fa-, oo) C p( A)

IIVr - Ill :S 2

for all 0

> 0.

and the operators Vr satisfy

< T < To.

Proof. By the formula

VT

= (A+ ti)(A- ti)- 1 =I- 2tR(t, A)

for t := 1.T we immediately obtain that

VT

ItR(t,A) = - 2 -.

(V.13) D

Then the proposition follows from the Hille-Yosida theorem.

Remark 2.9. The following nice property holds for cogenerators of C0 -semigroups

on Hilbert spaces: Contractivity of V automatically implies contractivity of every Vn T > 0. This follows from

IIVrxll 2 -llxll 2 = II(A + t!)(A- tit 1 xll 2 -llxll 2

= ((A+ t!)y, (A+ ti)y)- ((A- ti)y, (A- ti)y) = 4tRe (Ay, y)

for y := (A-ti)- 1 x. As we will see in Subsection 2.3, this property fails on Banach spaces. A result analogous to Proposition 2.8 holds for generators of bounded Co• semigroups as well. Proposition 2.10. For a densely defined operator A on a Banach space, the follow• ing assertions are equivalent.

(i) A generates a Co-semigroup

(T(t))t-?.o satisfying IIT(t)ll ::; M

for every t 2:: 0.

(ii) (O,oo) c p(A) and the operators Vr satisfy

II [VT;

Irll : ;

(iii) There exists To such that

II [VT;

M

for all T

>0

and n EN.

c;o, oo) C p(A) and the operators Vr satisfy

Irll : ;

M

for all 0 < T 0 such that ( oo) C p( A) and the operators Vr are con• tractive for every T E (0, To), then A generates a contractive Co-semigroup. (b) If there exists To > 0 such that (fa-, oo) C p( A) and the operators Vr satisfy IIVrnll ~ M for all T E (0, To) and n EN, then A generates a Co-semigroup (T(t))t20 with IIT(t)ll ~ M for all t ~ 0. Proof. Assertion (a) follows immediately from Proposition 2.8. To prove (b) as• sume that IIVrnll ~ M. Then we have

and (b) follows from Proposition 2.10.

D

Remark 2.12. In Proposition 2.8, Proposition 2.10 and Theorem 2.11 it suffices to consider {Vrn}~= 1 for a sequence Tn > 0 converging to zero. This again follows from the fact that in the Hille-Yosida theorem it suffices to check the resolvent condition only for a sequence {An}~= 1 C (0, oo) converging to infinity.

We finally observe the following. If V is contractive or power bounded, then so is the operator - V. Note that - V is the co generator of the semigroup generated by A - 1 if A -l generates a C0 -semigroup. However, contractivity or boundedness of (etA)t>o does not imply the same property of (etA- 1 )t>o (see Zwart [265] and also Subse~tion 2.3 for elementary examples). We refer to -Zwart [265, 266], Gomilko, Zwart [107], Gomilko, Zwart, Tomilov [108], de Laubenfels [60] for further infor• mation on this aspect. Remark 2.13. Assume that (0, oo) C p(A) and A- 1 exists as a densely defined operator. Then we have

So we see that contractivity (power boundedness) of Vr for all T > 0 or even for sequences Tn, 1 --+ 0 and Tn,2 --+ oo implies that A and A - 1 both generate a contractive (bounded) Co-semigroup. Conversely, Gomilko [105] and Guo, Zwart [113] showed for Hilbert spaces that if A and A - 1 both generate bounded semigroups, then the cogenerator V is power bounded (see Theorem 2.7) and hence so are all operators Vr by the rescaling argument.

Chapter V. Discrete vs. continuous

178

2.3 Examples In this subsection we discuss examples describing various situations on Banach spaces and show in particular that the Foia§-Sz.-Nagy theorem (see Subsection 1.3.1) fails even on 2-dimensional Banach spaces. In [108] Gomilko, Zwart and Tomilov show that for every p E [1, oo), p-=/:- 2, there exists a contractive operator V on[P such that (V- I)- 1 exists as a densely defined operator, but V is not the cogenerator of a C0 -semigroup. The idea of their construction is the following. Take the bounded operator A:= S1 - I, where St denotes the left shift given by St(x1, x2, x3, .. .) = (x2, X3, .. .). The cogenerator V of the semigroup generated by A is given by V = StR(2, St) and hence is contractive (use contractivity of St and the Neumann series for the resolvent). Further, one shows that A- 1 does not generate a C0 -semigroup which is the hard part. As a consequence one obtains that the contraction -Vis not the cogenerator of a Co-semigroup. Komatsu [152, pp. 343-344] showed that the operator A := Sr - I for the right shift Sr given by Sr(X1,X2,X3,···) = (O,x1,x2, ... ) on co satisfies the same properties, i.e., A - 1 does not generate a C0 -semigroup. Since the cogenerator V corresponding to A is contractive as well, we have a contraction on c0 which is not the cogenerator of a C0 -semigroup. The following example shows that even for X = C 2 the semigroup cogen• erated by a contraction need not be contractive. In particular, this example and Example 2.16 show that no implication in the Foia§-Sz.-Nagy theorem holds on two-dimensional Banach spaces. Example 2.14. Take X = C 2 endowed with II · liP, p -=/:- 2, and A := ( 01 (3 > 0. The semigroup generated by A is

_(e-t

T(t)-

0

e-2t)) ,

(3(ct _ e-2t

!2)

for

t 2': 0.

We first show that (T(t))t>O is not contractive for appropriate (3. Consider first p = oo and (3 > 1. We h;;:-ve IIT(t)ll = (1 + (3)e-t- (3e- 2t =: f(t). Since f(O) = 1 and f'(O) = (3- 1 > 0, the semigroup is not contractive. 1 Let now 2 < p < oo and define (3 := (3P - 1) 1i. Then

IIT(t) (~) [

= (e-tx

+ (3(e-t- e- 2t))P + e- 2pt =: fx(t)

for x

> 0.

We have fx(O) = xP + 1 = ll(x, l)IIP. Further, f~(O) = pxP- 1((3- x)- 2p, so the semigroup is not contractive if xP- ~ ((3 - x) > 2 for some x > 0. This is the case ._ for x .-

§_ 2.

Indeed, xp-1( (3- x ) -_

(!!.)p _ 3P-1 - ~ > 2. 2

We now show that the cogenerator V is contractive for (3 :S 3 if p = oo and 1 (3 := (3P- 1)1i if p E (2, oo). The cogenerator is given by

V =(I+ A)(A- I)-1 =

(~

!1) (-0~

=f)

=

(~ -~~).

179

2. Cogenerators

IIVII

= maxg, ~} ~ 1 for f3 ~ 3. For p E (2, oo) we have 1 = (-~'~)II~ = ({3~;;- ) ~ 1 if and only if f3 ~ (3P - 1) We see that for p E (2, oo] there exists a contraction such that the cogenerated semigroup is not contractive. The analogous assertion for p E [1, 2) follows by

So for p = oo we have

IIVIIP I

i.

duality. Remark 2.15. From Theorem 2.11, Remark 2.12 and the above example we see that there exist contractions V (even on C2 with [P- norm, p # 2) such that the operators V7 are not contractive for every T in a small interval (0, To).

The following example gives a class of contractive semigroups on (C 2 , ll·lloo) with non-contractive cogenerators. In particular, this provides a two-dimensional counterexample to the converse implication in the Foia§-Sz.-Nagy theorem.

Example 2.16. Every operator A generating a contractive Co-semigroup such that A - 1 generates a C0 -semigroup which is not contractive leads to an example of a contractive semigroup with non-contractive cogenerator. Indeed, by the previous remark, there exists T > 0 such that V7 is not contractive. Therefore, the operator T A generates a contractive semigroup with non-contractive cogenerator. For a concrete example consider X := C2 endowed with I · lloo and A as in Example 2.14. Then (etA)t>o is not contractive for f3 > 1. We show that the semigroup generated by A - 1 i~ contractive if and only if f3 ~ 2. Indeed, we have A- 1 = (

tA-l _ e -

(

-;=f) and

e-t 0

/3( e-t - e _.t.)) 2 t

e-2

'

t

~

0.

Therefore lletA- 1 IIoo =sup{ e-t + f3(e-~- e-t), e-~ }. Hence etA - l is contractive if and only if g(t) := e-t + f3(e-!- e-t) ~ 1 for every t > 0. We have g(O) = 1 and g'(t) = -e-t+ f3(e-t- ~e-~) = e-t[/3(1- ~d) -1]. Since the function t----+ 1- ~d is monotonically decreasing, we obtain that g' (t) ~ 0 for every t ~ 0 is equivalent to g'(O) ~ 0, i.e., f3 ~ 2. So we see that for 1 < f3 ~ 2 the semigroup generated by A - 1 is contractive while the semigroup generated by A is not contractive. The rescaling procedure described above leads to a contractive semigroup (generated by TA for some T) with non-contractive cogenerator. Zwart [265] gives another example of an operator A generating a contractive C0 -semigroup such that the semigroup generated by A - 1 is not contractive and not even bounded. He takes a nilpotent semigroup on X= Co[O, 1] such that the semigroup generated by A- 1 grows like t 114 . By the rescaling procedure we again obtain a contractive semigroup with non-contractive cogenerator. Remark 2.17. The above example for 2 < f3 ~ 3 yields a contractive cogenerator V such that the semigroups generated by both operators A and A- 1 are not contractive. So V is a contraction on (C 2 , I · lloo) such that operators Vr are not contractive for every T E (O,TI) U (T2,oo), 0 < T1 < 1 < T2, by Remark 2.12.

Chapter V. Discrete vs. continuous

180

2.4 Polynomial boundedness In this section, again following Eisner, Zwart [74], we show how polynomial bound• edness of a C0 -semigroup can be characterised by its cogenerator. We show first that polynomial boundedness of a C0 -semigroup on a Banach space implies polynomial boundedness of its cogenerator. This generalises results of Hersh and Kato [126] and Brenner and Thomee [41] on bounded semigroups. For the proof, which is analogous to the one of Theorem 2.4, see [74]. Theorem 2.18. LetT(·) be a Co-semigroup on a Banach space with cogenerator V. If IIT(t)ll S M(1 + tk) for some M and k E N U {0} and every t ~ 0, then IIVnll S Cnk+! some C and every n EN. Furthermore, this estimate cannot be improved, i.e., for every k E N U { 0} there exists a Banach space and a Co-semigroup satisfying IIT(t) II = O(tk) such that IIVnll ~ Cnk+! for some C > 0 and every n EN.

Our next result shows that for analytic semigroups the converse implication in Theorem 2.18 holds, i.e., polynomial boundedness of the cogenerator implies polynomial boundedness of the semigroup. Theorem 2.19. Let T(·) be an analytic C0 -semigroup on a Banach space with cogenerator V. If IIVnll S Cnk for some C, k ~ 0 and every n EN, then IIT(t)ll S M(1 + t 2 k+l) for some M and every t ~ 0.

Proof. Assume IIVnll S Cnk for some C, k ~ 0 and every n EN. Then r(V) S 1 and therefore >. E p(A) for Re >. > 0 by Proposition 1.3.3, where A denotes the generator ofT(·). Our aim is to show that there exist a0 , M > 0 such that

M

.

< Re >. < a0 ,

(V.15)

IIR(>., A) II S M for all >. with Re >. ;:::: ao.

(V.16)

IIR(>., A) II S (Re >.)k+l for all >. with 0

By Theorem III.l.20 this implies growth of IIT(t)ll at most as t 2k+ 1 for analytic semigroups. Take some a0 > max{O,w 0 (T)}. Then (V.16) automatically holds and we only have to show (V.15). Since T(·) is analytic, R(>.,A) is uniformly bounded on{>.: IIm.XI > bo, 0 < Re>. < a 0 } for some bo ~ 0. Moreover, R(>.,A) is also uniformly bounded on {>. : lim .XI S bo, ~ S Re >. < ao} as well. Take now >. with 0 < Re >. < ~ and -bo < Im>. < bo. By Proposition 1.3.3, where A denotes the generator ofT(·) we have IIR(>., A)ll

s 11:~~IIIIIR ( ~ ~ ~' v)

II·

(V.17)

By Theorem II.1.17, growth of IIVnlllike nk implies

6

.

IIR(J,t, V)ll S (I~LI- 1)k+l for all J.t w1th 1 < ifLI S 2

(V.18)

181

2. Cogenerators for some constant

C.

IJ.LI- 1 =

For

J.L :=

~~i and 0 < Re >.

< ~ we have

I>. + 11 - I>. - 11 4Re >. 1>.- 11 = 1>.- 11(1>. + 11 + 1>.- 11) < 1.

Then we use (V.18) to obtain

for

cl

:=

C(b6 + ~ )k+l.

So, by (V.17)'

C1(1 + IIVII) < C1(1 + IIVII) - I.A -1I(Re.A)k+l - 2(Re.A)k+ 1

I R(.A A)ll < '

which proves (V.15).

D

2.5 Strong stability We now discuss the relation between strong stability of a Co-semigroup and its cogenerator restricting our attention to Hilbert spaces. The following classical theorem is based on the dilation theory developed by Foia§ and Sz.-Nagy, see their monograph [238]. Theorem 2.20 (Foia§, Sz.-Nagy [238, Prop. 111.9.1]}. Let T(·) be a contraction

semigroup on a Hilbert space H with cogenerator V. Then lim

t----too

IIT(t)xll = n----too lim IIVnxll

holds for every x E H. In particular, T(·) is strongly stable if and only if its cogenerator V is strongly stable. Guo and Zwart obtained a (partial) generalisation of Theorem 2.20 to boun• ded C0 -semigroups. Theorem 2.21 (Guo, Zwart,

[113]}. LetT(·) be a bounded semigroup on a Hilbert

space H with power bounded cogenerator V. 1fT(·) is strongly stable, then so is

v.

Note that the assumption of power boundedness of V in the above theorem is satisfied if for example A - l exists and generates a bounded Co-semigroup as well, see Theorem 2. 7. No such result seems to be known on Banach spaces.

Chapter V. Discrete vs. continuous

182

2.6

Weak and almost weak stability

To conclude we look at the relation between weak and almost weak stability of a Co-semigroup and its cogenerator. We first show that for a large class of semigroups almost weak stability is preserved by the cogenerator. Proposition 2.22. LetT(·) be a relatively weakly compact C0 -semigroup on a Ba•

nach space such that its cogenerator V has relatively weakly compact orbits. Then T(·) is almost weakly stable if and only if V is. In particular, a bounded Co-semigroup on a reflexive Banach space with power bounded cogenerator is almost weakly stable if and only if its cogenerator is. Proof. Denote the generator and the cogenerator ofT(·) by A and V, respectively. By Proposition 1.3.3 we have

Pcr(A)

n ilR = 0

if and only if Pcr(V)

n r = 0.

Theorems II.4.1 and III.5.1 conclude the argument. The last assertion follows from Example 1.1.7 (a).

D

Open question 2.23. Let T(·) be a contractive C0 -semigroup on a Hilbert space. Is weak stability ofT(·) equivalent to weak stability of its cogenerator?

Remark 2.24. By Theorems II.3.13 and III.4.10, one can decompose every contrac• tion (semigroup) on a Hilbert space into a unitary and a completely non-unitary part, where the completely non-unitary part is always weakly stable. Moreover, a semigroup is contractive or unitary if and only if its cogenerator is (see Subsection 1.3.1). Therefore the question above restricts to the case of unitary groups and, by the considerations in Section IV.1, can be reformulated as follows. Assume J-L to be a probability measure on R Denote by v the image of J-L under the mapping iz + 1 JR-+f. iz - 1

ZH - - ,

Does the following equivalence hold: 11 is Rajchman on lR {::::::::} v is Rajchman on r?

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List of Symbols Au(T)

BUC(JR+) (C, 1) C(S) d

4

IDl FixT FixT(·)

FJ.L

r

kerq

£(H) sa C(H)+ £(X) Cu(X) fln

wo(T)

Pu(T) rgq

R(>.,T) Ru(T)

r(T) s(A) so(A) a

a(T) (T(t))t~o

T(·) WAP(S)

x-

approximate point spectrum of T .................... .......... 6 space of all bounded uniformly continuous functions on JR+ ... 81 Cesaro convergence .................... .................... ... 27 space of all continuous functions on S .................... ...... 8 (asymptotic) density .................... ................. 56, 114 (asymptotic) lower density .................... ................ 56 unit disc in C .................... .................... ......... 60 fixed space of T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 fixed space ofT(·) .................... .................... .... 20 Fourier transform of J.L ..•.....•.....•..... ...•.....•........• 136 unit circle in C .................... .................... ....... 10 kernel of q .................... .................... ............ 10 real vector space of all selfadjoint operators on H ............. 32 cone of all positive semidefinite operators on H ............... 32 space of all bounded linear operators on X .................... . 7 space C(X) endowed with weak operator topology ............. 7 nth Fourier coefficient of J.L .................... .............. 134 growth bound of the semigroup T(·) .................... ....... 6 point spectrum of T .................... .................... ... 6 range of q .................... .................... ............ 10 resolvent of T in >. .................... .................... ..... 6 residual spectrum of T .................... .................... . 6 spectral radius of T .................... .................... .... 6 spectral bound of A .................... .................... ... 6 pseudo-spectral bound of A .................... ................ 6 weak topology or weak operator topology, resp. . ............... 7 spectrum of T .................... .................... ......... 6 one-parameter semigroup of linear operators ................... 6 short notation for (T(t))t~o .................... ................ 6 space of all weakly almost periodic functions on S ............. 8 completion of X .................... .................... ...... 48

Index Co-semigroup almost weakly stable, 118 bounded, 79 induced, 142 mean ergodic, 20 polynomially bounded, 85 relatively compact, 13 weakly compact, 12 rigid, 149 along a set, 161 strongly stable, 100 uniformly exponentially stable, 91 uniformly mean ergodic, 22 weakly stable, 108 Blum-Hanson property, 56 bound growth, 6 pseudo-spectral, 6 spectral, 6 character, 10 cogenerator, 23 completely non-unitary contraction, 58 semigroup, 105, 111 density, 56, 114 lower, 56 function eigenvalue, 11 weakly almost periodic, 8 Haar measure, 10

ideal, 8 Koopman-von Neumann lemma 61 ' ' 115 Kreiss resolvent condition, 40 strong, 39 uniform, 40 Laplace inversion formula 28 ' Lyapunov equation, 75, 129 mean ergodic projection, 14, 20 means Abel, 15, 21 Cesaro, 14, 20 measure Rajchman, 135, 137, 182 rigid, 139, 141, 159 mixing mild, 156 strong, 144, 146 weak, 145, 146 operator almost weakly stable, 64 embeddable, 163 implemented, 34, 74 induced, 142 isometric limit, 50 mean ergodic, 14 polynomially bounded, 43, 51 positive, 33 positive semidefinite, 32 power bounded, 37 rigid, 148, 149 along a sequence, 161

Index

204 strongly stable, 45 uniformly exponentially stable, 38 uniformly mean ergodic, 18 weakly stable, 55 with relatively weakly compact orbits, 12 with p-integrable resolvent, 87 relatively dense set, 55, 109 Ritt resolvent condition, 40 semiflow measure preserving, 141 rigid, 159 semigroup abstract, 8 compact, 8 implemented, 35, 129 semitopological, 8 structure of compact, 9 spectrum approximate point, 6 residual, 6 strong* operator topology, 68 syndetic set, 55, 109 theorem Arendt-Batty-Lyubich-Vu, 51, 104 Datko-Pazy, 95 Eberlein-Smulian, 5 Foguel, 58, 111 Foia§-Sz.-Nagy, 25, 52, 57, 110, 181 Gearhart, 97 Gelfand, 49 Gomilko-Shi-Feng, 82 Guo-Zwart, 175, 181 Jacobs-Glicksberg-de Leeuw abstract version, 10 for C0-semigroups, 13, 119 for operators, 12, 65 Jamison, 38, 80 Jones-Lin, 64 Katznelson-Tzafriri, 50

Krein-Smulian, 6 Muller, 47, 57 mean ergodic, 16, 22 Rolewicz, 96 spectral mapping, 92 circular, 94 weak, 93 Tomilov, 106 van Neerven, 95 Weiss, 96 Wold, 71, 112 transformation measure preserving, 141 rigid, 154 unilateral shift, 71, 108